Jakob Pelletier and Mikael Backlund led the way for the Flames with a goal and an assist each, while Dan Vladar - who got the start on back-to-back nights - made 25 saves.
Jonathan Huberdeau polished off Calgary's other marker.
The Flames got caught on a bad change in OT and Jack Eichel took full advantage, re-grouping in the neutral zone before setting up Jonathan Marchessault for a clean look off the rush. His attempt rang off the crossbar, but Pietrangelo bowled his way to the blue paint and casually deposited the loose change for the winner.
A sharp-angle shot from Zach Whitecloud made it a one-goal game only 5:08 into the final frame, before William Carrier tied the game off a mad scramble in front with 7:24 to play.
The Flames were out-shot 9-1 in the third, but despite that and surrendering the two Vegas markers, the Flames had a chance to win it on a late powerplay but Rasmus Andersson's point shot clanked off the crossbar to force 3-on-3.
The powerplay - which looked dangerous all night - finished 1-for-4.
With the OT loss, the Flames fall to 27-20-12 on the year.
They close out this three-game road swing on Saturday against the defending Cup champion Colorado Avalanche.
"We need these points," Huberdeau said. "We're up 3-1 in a game and in the third period, they come back. We know OT hasn't been our greatest strength this year, so we've got to find a way, somehow, and keep that lead. I feel we got the lead, 3-1, and stopped playing offence. We can't do that. We've got to keep playing and score the fourth goal so we get a bigger lead.
"We had a lot of shots in the first, created some chances. But in the third, we had, what, one shot? That's not our team. We let these guys come at us. ... I think we've got to keep the pedal down. Obviously, you're going to be a bit tired in the third. But if you want to make the playoffs, we need these points."
The Flames were all over the Knights early and after stringing together a series of bold, battering shifts deep in enemy territory, they earned the game's first powerplay when former Flame Byron Froese was flagged for hooking Blake Coleman.
The visitors threw everything but the kitchen sink at Vegas netminder Laurent Brossoit, including a sizzling snapshot from Tyler Toffoli, a wrister from Elias Lindholm, and a pair of one-timers off the sticks of Andrew Mangiapane and Dillon Dube, but the former Flames draft pick stood tall in only his second NHL game in the last calendar year.
But Brossoit, simply, could not stop them all.
Less than 24 hours after shelling the Arizona Coyotes with a season-high 51 shots, the Flames continue to pile on the rubber - and on their 16th of a 19-shot opening stanza - they finally hit paydirt.
With Shea Theodore off serving holding infraction, Pelletier banged home a loose puck to score his third of the season and second in as many games. Mikael Backlund picked up the primary assist, as he teed off from the right circle, allowing Pelletier to swoop in and cash in from close range.
The goal came at 13:25, with Noah Hanifin bagging the other helper.