Colorado outshot Vegas 11-5 in the second period, with Fleury saving 10 of them.
Joonas Donskoi scored on a one-timer from the slot at 16:28 to make it 2-0. It ended a shift that lasted 2:12 in the Vegas defensive zone.
"Fleury made some big saves, especially in the second period," Golden Knights forward Alex Tuch said. "We wanted to help him out. He's bailed us out all year long, and we knew he wanted that first goal back."
Tuch scored 1:03 into the third period to make it 2-1, and Jonathan Marchessault tied it 2-2 at 4:07.
Stone scored at 50 seconds to end the second overtime game in the best-of-7 series. Colorado won Game 2 on Mikko Rantanen's goal at 2:07.
The Avalanche almost scored 10 seconds into overtime Tuesday, but Fleury saved shots by Gabriel Landeskog and J.T. Compher on the first shift.
Then, after Max Pacioretty blocked consecutive shots by Avalanche defenseman Ryan Graves, Stone received an outlet pass from Pacioretty and scored on a breakaway to give Vegas the lead in the best-of-7 series.
"I think 20 seconds into overtime, the game should've been over," Stone said. "We should've been going home, but 'Flower' made a huge save and kept us in it, and gave me that opportunity to end it."
Fleury (88-67) tied Ed Belfour and Billy Smith for fourth in Stanley Cup Playoff wins, behind Patrick Roy (151), Martin Brodeur (113) and Grant Fuhr (93).
One more will put the Golden Knights into the third round for the third time in their four-season history. Game 6 is at Vegas on Thursday (9 p.m. ET; NBCSN, CBC, SN, TVAS).
"He's bailed us out a number of occasions this year when we've made mistakes," Golden Knights coach Peter DeBoer said. "I had no doubt after that [goal] that he was going to be lights out and our group was going to be lights out for him."