In Britain, where she was born, Mirza has overcome the prejudices of both Muslims and non-Muslims to become a leading figure on the stand-up comedy circuit.
And far from hiding her religion, she makes a point of emphasizing it by the clothes that she wears whenever she goes on stage.
Muslim women traditionally wear the hijab or headscarf as a sign of modesty. It's supposed to make them anonymous. But for Mirza, it's become her trademark, and it's made her famous.
It's what this 27-year-old unmarried Muslim wears, not just when she goes to the mosque, but also when she performs in the male-dominated, beer-swilling world of British stand-up comedy. Correspondent Ed Bradley reports.
With her dry wit and deadpan delivery, Mirza doesn't spare herself or her predominantly white audiences:
"I'm really pleased to be here tonight actually because my dad has let me out for the night. So I'm not going to stay long. Actually, he's picking me up in 10 minutes. He thinks this is a library."
"Now, I don't drink alcohol. It's against my religion to drink alcohol, but my English friends, they always seem to have such a great time when they go out. They get drunk. They wake up in the morning and say, 'I had a brilliant time last night. Where did we go again? Who is the father of my child?' I want to be drunk, I want to cry, fall over and roll down the street. It looks like so much fun."
Mirza says that on stage, she just tells the truth about how she sees and feels about the world: "I tell the truth about my experiences and my life."
She once said that she didn't make jokes about sex because, as a devout Muslim woman, she didn't know about that. "I never make jokes about sex because I've never had it, and that's true," says Mirza. "I've never actually gone on stage and made up a story just because I thought it was funny. There would be no point in me being a stand-up if that was the case."
Born in Birmingham, England, to first generation Pakistani immigrants, Mirza says that from a young age she knew what she wanted to do with her life. But it wasn't a career choice that went down well in the community where she was raised.
"All the adults would ask the kids, 'What do you want to be when you grow up? What do you want to be?' And all the kids were really well-trained. They'd all say, 'Well, I want to be a brain surgeon, I want to be a dentist, I want to help all the sick kids in Africa,'" recalls Mirza.
"And when they'd say to me, 'What do you want to be?' I'd say, 'I want to be an actress, I want to be on the stage.' And my dad would go, 'Leave the room. You've showed us up, leave the room.' And it was really seen as a shameful thing to say in front of people in the community, 'I want to be on the stage.'"
As a good daughter, Mirza followed her parents' wishes, and after earning a degree in biochemistry, she became a teacher in London. But in the evenings, she secretly enrolled in drama school, and was soon making a name for herself on the stand-up comedy circuit in the city's pubs and clubs.
In the summer of 2001, she even played that Mecca of British entertainment, the London Palladium. But then came the event that threatened to put an end to her career before it had barely begun.
"When 9/11 happened, I thought I'm never going be able to do stand-up comedy ever again. Nobody's going to want to hear comedy from a Muslim woman after 9/11," says Mirza. "There was so much hatred towards Muslims that I just thought, I was scared to go on stage."
But three weeks later, in a club in London's Soho, Mirza decided to confront the problem head on: "I went on stage and said, 'Hello, my name's Shazia Mirza, at least that's what it says on my pilot's license,'" says Mirza.
"I remember people gasping, going, 'Ah!' So they were really shocked and then they laughed. There was so much tension in that room when I'd walked on that people were actually quite relieved that they could laugh."
The pilot's license joke went around the world and was translated into a dozen languages. Mirza was asked to perform in Europe, Japan and in the United States. She even used that line in her act when she came to America.
"Most of the people in the audience laughed, but there was an American man on the front row, sat right in front of me and he just put his head in his hands like that and he started shaking his head," recalls Mirza.
"And I saw him afterwards and he said, 'My sister died in the World Trade Center.' And I said to him, 'I'm not making jokes about the people that died. I condemn what happened, but what I'm trying to say is that not all Muslims are terrorists.'"
But not all Muslims appreciate Mirza's attempt to defuse religious and cultural tensions. Some send her vicious emails calling her a prostitute and a disgrace to her religion. And when she came to London's Brick Lane neighborhood, to take part in an Asian comedy show, the conservative Muslim community who live here certainly didn't see the joke.
"I had only done the first two lines of my act and there was a group of Bangladeshi boys in the audience. They jumped up onto the stage and they grabbed me by the neck and they pushed me off the stage," says Mirza.
Did she consider giving up stand-up comedy because of that? "Yeah, I was so scared, actually, that I was scared to go home that night," recalls Mirza.
"For about two weeks afterwards, I was scared to be in the house by myself because I thought that they would come after me and get me. But then, I started doing whatever I wanted to do."
And Mirza did whatever she wanted to do when she appeared last summer at the Edinburgh International Arts Festival.
"Now, are there any Muslims in here tonight," said Mirza in front of a multi-cultural audience. She then joked about a pilgrimage she made to Islam's holiest site.
"Last year, I went to Mecca to repent my sins, and I had to walk around the black stone. All the women were dressed in black, you could only see their eyes. And I felt a hand touch my bottom. I ignored it. I thought, 'I'm in Mecca, it must be the hand of God.' But then it happened again. I didn't complain. Clearly, my prayers had been answered."
Does Mirza think that there's anything in the Koran that prevents her from being a stand-up comedian?
"No, I don't. I don't believe that what I'm doing is wrong at all. Islam gives women a lot of power, and it gives women a lot of freedom as well," says Mirza.
"And I'm sure that, you know, Prophet Mohamed, peace be upon him, he laughed, everybody laughed. I mean, how are you gonna get through life if you don't laugh? … I'm sure that God has a sense of humor. I hope he finds my jokes funny."
But there was one subject that Mirza found no laughing matter, and that was telling her parents about her new career and the world in which she now lived.
What did she do? She didn't tell them at all. In fact, the first inkling her parents had that their daughter was no longer a teacher was when they saw her on a TV comedy quiz show.
"And my dad said, 'I'm so proud of you. You was on the game show last night,' and I don't think he quite realized that actually I was a stand-up comedian, and I didn't just happen to arrive on this game show," recalls Mirza. "Then I started to appear on TV more and more, and I think it suddenly began to dawn on him that there's something not right here. How can she be a teacher and be appearing on the TV at the same time? And I think for a long time, he was in denial about it."
So when 60 Minutes invited ourselves to the Mirza family home in Birmingham for a Sunday lunch party, we were surprised to find that Mirza's father, Ayaz, now claims to be one of his daughter's greatest fans.
"Well, we really feel quite privileged really," says Ayaz Mirza. "I'm glad she hasn't gone into any other profession." Then, he says to his daughter, "So, you must give me credit that I have readily accepted the change."
As for Mirza's mother, Sarwat, her main regret is that her daughter hasn't yet found a husband. And it's a situation that makes her the butt of many of Mirza's jokes.
"Single, Muslim men in Britain. And in it, it has height, weight and size of beard -- all the most important things in a marriage. But my mum's so desperate now, she says, 'Look, get a white man and convert him.'"
Mirza says her mother has a "massive" list of good Muslim men for her: "Initially, she used to tell the boys that I was a teacher, and I used to get loads of offers for marriage. And it's funny, because now when she does tell boys and their parents that I'm a comedian, they don't even get to meet me. They say, 'We're not interested.'"
Mirza says, however, that her husband would have to be a Muslim. The only exception? "I would only marry George Clooney if he converted to Islam," says Mirza. "That'd be all right."
Mirza's mother doesn't know who George Clooney is until Shazia tells her he's a Hollywood film star. "Oh, right," says Sarwat Mirza. "Well, I don't mind."
"You're desperate now," says Shazia, laughing. "You don't care. Anything will do."
But after everything she's been through to get this far, the difficulty of finding a husband doesn't deter Mirza from the career that she's always dreamed about.
Drawing on her experiences as a single Muslim woman traveling around the world, she continues to challenge the prejudices and preconceptions of her audience on stage:
"I got on the plane to Denmark dressed like this, and this woman refused to sit next to me. So I said to her, 'I'm going to sit on this plane and blow it up. And you think you're going to be safer three rows back?'"
"I do want to do something different. I want to do something challenging and I want to take risks, but above all I want people to be entertained," says Mirza. "I want them to laugh. Because the only way of talking about some issues sometimes is through humor."