Elliott was born in Tucson, Arizona and was the youngest of three boys. He attended the G.A.T.E. (Gifted and Talented Education) program at Tolson Elementary School in Tucson. He played high school basketball at Cholla High School (now Cholla High Magnet School) on Tucson's west side. After graduating in 1985, he remained in Tucson to play college basketball at the University of Arizona. Under the tutelage of Lute Olson, Elliott was selected as a consensus all-American during his junior and senior years, and led the Wildcats to the Final Four in his junior year (1988). Elliott was the first player to break Lew Alcindor's (Kareem Abdul-Jabbar) all time PAC 10 career scoring record. After an exceptional senior season, Elliott won the Wooden Award. He is still the University of Arizona's all-time leading scorer.
Elliott was expected to be traded to the Houston Rockets in exchange for Robert Horry during the 1994-95 season, however the trade fell through when Elliott failed his physical. Robert Horry went on to win the team's second NBA Championship with the Rockets later that season.
Elliott was an instrumental part of the Spurs' successful NBA title quest in 1999. In Game 2 of the 1999 Western Conference Finals, he hit a 21-foot 3-point shot against the Portland Trail Blazers, giving the Spurs a 1-point lead with 9 seconds left to play in regulation. The shot was called "Memorial Day Miracle" because of its improbability and the date on which it was made. The pass was nearly stolen by Blazer Stacey Augmon, and Elliott caught the ball within an inch of the sideline (narrowly avoiding going out of bounds), and had to stay on his tiptoes rather than planting his feet. When Elliot released the ball, it just avoided the outstretched arms of 6'11" opponent Rasheed Wallace. This play shifted the momentum of the series to the Spurs.
Shortly after the championship run, Elliott announced that he had played despite having a kidney disease, focal segmental glomerulosclerosis, and that he would require a transplant. He underwent surgery on August 16 of that year, receiving a kidney from his older brother, Noel. On March 13, 2000, Elliott became the first player to return after a kidney transplant, in a game against the Atlanta Hawks. He announced his retirement in 2001.
He finished his career averaging 14.2 points per game, 4.3 rebounds per game and 2.6 assists per game. Elliott is the third all-time franchise leader in three-point field goals made (563) and attempted (1,485). He is also the only player in Spurs history to rank among the franchise's top ten in six different statistical categories: games played (third, 669), points (fourth, 9,659), rebounds (sixth, 2,941), assists (seventh, 1,700), steals (eighth, 522), and blocks (ninth, 257).
After retiring, Elliott became a basketball analyst for The NBA on NBC and, during the 2003–2004 season, for ABC Sports and ESPN. He left that position for the 2004–2005 season and became the color commentator for the Spurs' local broadcasting. On January 5, 2013, he joined Fox Sports 1, calling his first college basketball game with the network.
On March 6, 2005, his #32 jersey was retired and hung in the rafters of the AT&T Center. His #32 jersey is also retired by the University of Arizona.