Michael Amadio scored shortly after at 9:59 to cut it to 4-3; Stone scored his second goal of the game at 14:47 to tie it 4-4; and Max Pacioretty scored on a rebound at 15:54 for the game-winner and 5-4 final.
Jason Robertson had a goal and an assist to extend his point streak to seven games (five goals, six assists) for the Stars (13-8-2) in their first loss since Nov. 18. Braden Holtby made 39 saves.
"That's not our game," Stars coach Rick Bowness said. "We've got to get back to playing our game tomorrow night [at the Los Angeles Kings]."
The Golden Knights (15-10-0) won their third straight despite going 0-for-3 on the power play in the first period. Jonathan Marchessault scored in his 300th game with the Golden Knights, who made their League-leading fourth multi-goal comeback of the season.
Stone also had an assist in extending his point streak to six games (two goals, five assists. Pacioretty had two points to extend his streak to six games (five goals, four assists) since returning from a lower-body injury that kept him out for 17 games from Oct. 20-Nov. 22.
"I think last game was the way we wanted to play as well. Playing a good team in [the] Calgary [Flames on Sunday, a 3-2 win], knew that it wasn't going to be an easy one," Stone said. "But coming back, battling that adversity was huge for our team and hopefully can kickstart something big finishing off this homestand.
Robin Lehner was pulled in the second period after allowing three power-play goals on 10 shots. Laurent Brossoit made 12 saves in relief.
Coach Peter DeBoer said the call to change goalies was made for the entire team.
"You feel for the goalie because you can't hang that all on him," DeBoer said. "That's a last resort for me; I don't pull goalies very often. I just felt it was the right things to do tonight."
Robertson gave the Stars a 1-0 lead on the power play at 9:30 of the first period.
Stone tied it 1-1 19 seconds into the second period seconds after the Golden Knights' power play expired.
Seguin gave the Stars the lead back, 2-1, four seconds into the power play at 2:05 of the second period.
Pavelski made it 3-1 at 7:42 of the second off a deflection on a point shot from John Klingberg, the Stars' third power-play goal of the game. Dallas finished 3-for-3 on the man advantage.
"Special teams obviously were huge. Goaltending was great; 5-on-5, we just didn't have the energy that we normally play with," Bowness said. "Simple as that."