Devin Harris, who had 21 points and nine assists, hit a 3-pointer with 1.2 seconds to play. Kirilenko stole the inbound pass under the basket but lost control as he went up for a shot.
"My brain stopped working," Kenyon Martin admitted after the game. Replays showed Martin, who was still out of bounds, touched the ball after Kirilenko stole it from him, but the referees didn't review it.
"We were just so happy to run off this court with a win, because we haven't been able to do that here very often over the years," Lawson said.
While the Nuggets have thrived since trading their All-Star, the Jazz have sunk since dealing theirs.
The Jazz have lost four of five since they traded star point guard Deron Williams to New Jersey. Utah dropped two games behind Memphis for the eighth and final slot in the Western Conference.
Utah also has lost seven straight at home -- the team's worst home skid since 1982 -- and is 1-6 under new coach Ty Corbin.
Against the most foul-prone team in the league, the Nuggets repeatedly took the ball to the basket down the stretch. Denver made 15 of 25 free throws in the fourth quarter alone. The Jazz attempted 12 free throws the entire game.
"We were attacking on screen-and-rolls the whole fourth quarter," Lawson said. "They were in the bonus early so I was determined to get it to the basket and get to the line, because free throws are the easiest way to score."
The Nuggets are the top-scoring team in the league at 107.5 points per game and have not dropped off offensively since the trade. But the new-look Nuggets also have clamped down on the defensive end.
Although the Jazz shot 52 percent, the Nuggets guarded best when the game was on the line.
"On defense, we got the stops when we needed to. That was the key," Afflalo said.
After Harris made a 3-pointer for Utah's last lead at 96-94, the Nuggets forced three straight turnovers.
"Big win, exactly what we needed," Denver coach George Karl said.
The Jazz led 90-85 on Al Jefferson's spinning layup, but the Nuggets scored nine straight points to make 94-90 with 3:45 remaining in the game.
"We got up five points, and then we got away from what we were doing, took a couple bad shots and turned the ball over," Corbin said.
Martin hit back-to-back jumpers to put Denver up 74-66 with 3:09 left in the third quarter. But the Jazz, sparked by the fast-breaking Harris, scored 10 straight points and took the first lead of the game 76-74 heading into the final period.
The crowd gave the Jazz an extended standing ovation at many points in the fourth quarter but stood in disbelief and collectively groaned as Kirilenko's final shot went awry.
"[Martin's] hand was on the ball. It wasn't a foul but I didn't really get the ball and I tried to finish without it," Kirilenko said.
Hitting seven of their first eight shots, the Nuggets burst out to an early advantage and did not relinquish the lead until late in the third quarter.
Newly acquired Derrick Favors (cold) and reserve guard Ronnie Price (sprained toe) did not dress for Utah. Corbin, who assumed head coach duties without a contract since Feb. 10 when Jerry Sloan abruptly resigned, finally signed a multiyear deal with Jazz on Tuesday. ... Jazz G Earl Watson had to leave the game with 6:30 remaining to close a cut to his face. ... Neither team led by more than nine points.