The Jazz led by as many as 21 points in the second quarter, 18 in the third and 15 in the fourth before Philadelphia rallied. The Sixers had a 103-101 lead with 11.5 seconds left in regulation thanks to a 14-point quarter by Andre Iguodala.
But C.J. Miles' driving layup tied it and Iguodala missed what would have been the game-winner.
Kirilenko then took over, hitting a 3-pointer, putback dunk and two free throws to seal the win.
The Sixers haven't won in Utah since Jan. 5, 2005, losing 10 of the last 12 overall against the Jazz.
Al Jefferson led the Jazz with 30 points and 17 rebounds, including 11 defensive boards.
Rookie forward Derrick Favors, making his first start in place of the injured Paul Millsap, added 11 points, six boards, two assists and a steal for the Jazz.
Iguodala led Philadelphia with 23 points, while reserves Lou Williams added 22 and Thaddeus Young 10. Elton Brand started slowly for the Sixers, hitting just 1 of 7 shots in the first half before finishing with 19. Jodie Meeks added 17 as five Philadelphia players scored in double figures.
It wasn't enough against Utah.
Jazz coach Ty Corbin started Favors to give Utah a defensive boost after seeing his team give up 39 points in the first quarter against Minnesota on Friday and 37 Saturday night at Chicago. The Jazz ended up losing those two games by a combined 39 points.
Favors started strong with three rebounds, two assists, two points and a steal in the first six minutes, but he picked up two quick fouls.
Still, the Jazz led 28-23 after one, and 60-42 at halftime. The Jazz were without Millsap for the fourth straight game because of a knee injury. Reserves Ronnie Price, Mehmet Okur and Francisco Elson continued to be inactive, while rookie Gordon Hayward was in street clothes Monday night because of a foot injury.
Guard Raja Bell did not start because of a toe injury but he entered in the second quarter and contributed five points, four rebounds, and one assist.
In the fourth quarter, the Jazz also lost starting point guard Devin Harris, who left with a hamstring injury and did not return.
Notes: Jazz coach Ty Corbin said X-rays taken Monday morning on Hayward's left foot were negative for a stress fracture. Gordon will continue to receive treatment. ... The Sixers scored just 19 points in the second quarter.