Kelly agreed to a five-year contract worth $23.3 million with a $9 million reciprocal buyout, according to ESPN.
"I am thrilled to welcome Chip Kelly to Westwood," UCLA athletic director Dan Guerrero said in a statement. "His success speaks for itself, but more than that, I firmly believe that his passion for the game and his innovative approach to coaching student-athletes make him the perfect fit for our program. 'Champions Made Here' is more than just a mantra at UCLA, and I'm confident that Chip will lead UCLA Football back to competing for championships."
School officials are expected to introduce Kelly during a news conference on campus early next week.
Kelly also had been considering the vacant job with the Florida Gators.
UCLA fired Mora last Sunday, a day after a third straight loss to rival USC. Mora was 46-30 in six seasons with the Bruins.
Kelly last coached collegiately at Oregon, going 46-7 with the Ducks from 2009 to 2012. He guided Oregon to the national championship game after the 2010 season as well as three conference titles in his four years at the school.
Kelly, currently an analyst at ESPN who turned 54 Saturday, went 46-7 at Oregon, which averaged 44.7 points a game during that span. His final team in 2012 went 12-1 and averaged 49.6 points per game.
Kelly left Oregon in 2013 for the NFL shortly before NCAA sanctions were handed down at the school. Kelly was handed an 18-month show-cause penalty, which expired in late 2014.
Kelly was hired by the Eagles in 2013 and went 10-6 in each of his first two seasons in Philadelphia before being fired after a 6-9 campaign in 2015. The San Francisco 49ers then hired Kelly, but he lasted just one season, going 2-14 before being fired in January with three years remaining on his contract. He was 28-35 in four seasons as an NFL coach.