"I feel really blessed and really lucky," Jones said after winning the album of the year at the end of the 45th annual awards show.
Jones, who won in every category where she was nominated, tied Lauryn Hill and Alicia Keys for most wins by a female artist in a single night.
As unknown as triple-winner Bruce Springsteen was acclaimed before the last year, Jones won five individual Grammys.
"I never ever thought that the music I made would become popular music, so this is amazing," Jones said as she picked up an award for best pop vocal album.
She also won for best female pop vocal, best new artist and for record of the year for "Don't Know Why," written by Jesse Harris, who won the song of the year Grammy.
Her album, "Come Away With Me," earned best engineered album and producer of the year honors for Arif Mardin.
The 23-year-old New Yorker, who emerged last year after signing with the tiny Blue Note label, sold more than 6 million copies of her debut worldwide. Jones, the daughter of Indian sitar master Ravi Shankar and New York concert producer Sue Jones, appeared almost overwhelmed in accepting one award from the "Queen of Soul," Aretha Franklin.
"I can't believe this," she said breathlessly.
Harris, who thanked Jones for "being a pal," said backstage that he "thought for certain Bruce was going to win."
Springsteen was clearly a favorite of the crowd inside Madison Square Garden as the Grammys returned to New York for the first time in five years. His Sept. 11-inspired "The Rising" was honored for male rock vocal, rock album and rock song, although the Boss lost out for album and song of the year.
Performing only a few miles north of ground zero, Springsteen delivered a rousing performance of the title track midway through the show as the crowd screamed, "Bruuuuce!!!!"
Later, Springsteen joined Elvis Costello, Little Steven Van Zandt, Dave Grohl and Tony Kanal for a rousing tribute to the late Joe Strummer - a version of the Clash's "London Calling."
The Dixie Chicks, after a bitter legal battle with their label Sony, took best country album among their three Grammys. The trio's multi-platinum "Home," a return to their country roots, was co-produced by member Natalie Maines' father, Lloyd.
"I want to check the record books and see how many fathers and daughters have won Grammys together," she said, grabbing her father.
Among the double winners were Eminem, Coldplay, India.Arie and Nelly. Eminem's second Grammy came for best rap album, the third time he's won in the category.
Instead of offering an acceptance speech, Eminem rattled off a list of rappers who had inspired him, including Run-DMC, LL Cool J, Tupac Shakur and the Notorious B.I.G.
"Thank you, because I learned from all of you," he said.
Eight artists were nominated for five Grammys apiece, with all but teen rocker Avril Lavigne going home winners. Perennial Grammy favorite Sheryl Crow, neosoul singer-songwriter Raphael Saadiq and R&B newcomer Ashanti were single winners.
Country singer Alan Jackson, who wrote "Where Were You (When the World Stopped Turning)" in the weeks after the terrorist attacks, won best country song for the mournful tune.
India.Arie, who was nominated for seven Grammys last year but lost them all, finally got her first two, winning for best urban-alternative performance for "Little Things" and best R&B album for "Voyage to India."
Among the other winners: the previously unheralded Funk Brothers. The groundbreaking house band for Motown Records, the focus of the recent documentary "Standing in the Shadows of Motown," won two trophies.
Some other veteran artists added to bulging trophy cases: bluesman B.B. King won two, for 13 in his career, while Johnny Cash won his 11th and Tony Bennett his 10th - while soul legend Solomon Burke won his first.
"We got a Grammy, baby!" Burke said as he hoisted his Grammy.
The disc "Vaughn Williams: A Sea Symphony" garnered three awards, including best classical album.
The show opened with Dustin Hoffman - one in a revolving series of New York-based hosts - introducing a reunion performance by lifetime achievement award winners Simon and Garfunkel. The pair, who sang "The Sound of Silence," have shared a tumultuous relationship; this was their first performance together in a decade.
Hoffman, before exiting, provided two gaffes. He mispronounced the Boss' name as "Springstreet," and introduced No Doubt by singing "Say Baby" - their hit is "Hey Baby."