(CBS News) Tea Party-backed Republican Ted Cruz made history Tuesday night by becoming the first Hispanic to win a Senate seat in Texas.
He will be the third Latino in the Senate, joining Bob Menendez, D-N.J., and Marco Rubio, R-Fla. Tuesday night, he won the Texas seat held by retiring GOP Sen. Kay Bailey Hutchison, defeating Democratic challenger Paul Sadler, 57 percent to 41 percent.
On "CBS This Morning," Cruz said Republican candidates need to do a better job of making their case with Hispanics. He said, "There is no doubt that both parties competed for the Hispanic vote and at least last night President Obama earned much more of the Hispanic vote than did Mitt Romney.
"At the end of the day, Republicans need to do a better job at laying out the key to economic opportunity is economic growth and is jobs and that the policies that produce that, I think, are not the policies of this administration. At least last night, that was not the result nationally."
Cruz suggested the Republicans' issue with reaching Hispanics was articulating their message effectively. "I think the values in the Hispanic community are fundamentally conservative," he said. "But you've got to have candidates that connect with that community in a real and genuine way and communicate that the values between the candidate and the community are one and the same."
Asked by "CBS This Morning" co-host Norah O'Donnell about immigration and whether he would support a measure akin to Florida Sen. Marco Rubio's proposal that would allow a pathway to citizenship such as visas for illegal immigrants who are already in this country, Cruz said, "I don't think the answer to our immigration problems is amnesty."
Pressed that Rubio has said it's not amnesty he wants to give, but rather a visa which is not amnesty, Cruz said, "Well, look, it would depend on what a particular proposal is, so until I was looking at a particular proposal I couldn't tell you. What I can tell you is my overall principles and how I will analyze any of these issues, which that I think immigration is an issue where there is widespread agreement among Americans.
"I think most Americans agree, number one, that we need to get serious about securing the border and stopping the problem of illegal immigration and number two most Americans agree that we should remain a nation that celebrates legal immigrants, Americans by choice, is what Ronald Reagan said, and we need to improve our legal immigration system."
Asked by O'Donnell who is now not celebrating legal immigration, Cruz said, "You talked about the tone. Sometimes the tone in which immigration is discussed doesn't do that. I'll tell you in my campaign, I spent every day talking about my father's journey 55 years ago coming from Cuba with nothing. With $100 in his underwear, washing dishes to pay his way through college. That's the American dream. That's the American story and we need to improve the legal immigration system to streamline it and make it more effective."
For more with Cruz, including talk about what issues he would like to work with President Obama on in the next four years, watch the video above.