Staal beat Mike Smith with a wrist shot from the circle 4:02 into overtime. Montreal improved to 2-9 in overtime games. Edmonton fell to 2-2.
''I just love that we stayed with it and competed,'' Staal said. ''Tonight was a lot of fun to be part of. I could feel the energy through the TV screens. I wish it was full because I know what this place is like when it's full, especially in that type of moment. But it was awesome nonetheless.''
Staal played 16 minutes 56 seconds, and won 14 of 18 faceoffs.
Edmonton was outplayed early but led 1-0 after the first and 2-0 after the second.
Devin Shore and Darnell Nurse scored for Edmonton, which managed goals in the last minute of both the first and second periods.
Josh Anderson and Tomas Tatar scored for Montreal in its third-period comeback.
After some fine work along the boards, Corey Perry found Anderson parked at the crease 1:45 into the third period to get the Habs back in the game. It was Anderson's 14th of the season
Tatar tied it up less than four minutes later, snapping a shot past Smith for his eighth after Paul Byron took advantage of an Edmonton giveaway.
''Best way to protect a lead is to play in the other team's end and we were just the first five minutes or so (of the third) too passive, didn't do enough forechecking to give us the momentum we needed,'' Edmonton coach Dave Tippett said.
Tippett gave credit to Staal.
''He made a great shot. Smitty was great all game but he made a great shot to end the game,'' he said.
The Canadiens lost forward Brendan Gallagher in the first period after being hit in the right hand by an Alexander Romanov shot while screening Smith. He did not return.
Montreal held a 15-5 edge in shots after the first period, an advantage narrowed to 22-19 after the second. It was 29-22 for the Canadiens after regulation.
The Oilers (23-14-2) were coming off a 3-2 win over Calgary on Friday. Their game Saturday against Vancouver was postponed due to the Canucks' COVID-19 outbreak.
Montreal (17-9-9) saw its three-game win streak snapped in a 6-3 loss to visiting Ottawa on Saturday.
The Canadiens had won three of the four previous meetings with Edmonton this season, blanking the Oilers 4-0 last time out March 30 - a game in which Edmonton recorded just 17 shots and Carey Price earned career shutout No. 49.
Smith had the shutout Feb. 11, when Edmonton beat Montreal 3-0.
Staal played on a line with Jonathan Drouin and Tyler Toffoli. It was regular-season game No. 1,273 for the 36-year-old who came into the contest with 439 goals and 592 assists.
''I've been lucky to play a long time in this league,'' Staal said. ''I'm grateful every day. Every day to be able to be out there and compete and play, it's awesome. And there's nothing better.
''So this was a moment for sure that we'll keep in the memory bank but I'm hoping there's even bigger and better ones here in the next while.''
Montreal acquired the native of Thunder Bay, Ontario, from Buffalo on March 26 in exchange for a third- and fifth-round pick in the 2021 draft. He had to undergo a one-week quarantine before suiting up.
Staal saw power-play time, but negated a second-period man-advantage by taking a hooking penalty 12 seconds later.
Smith was the busier goalie early, with the Canadiens holding a 12-1 edge in shots through the first 14 minutes.
Smith had to be sharp late in the period, stopping a Toffoli tip and then Drouin on the rebound.
But it was Edmonton that struck first through its fourth line and fifth shot, with 21 seconds remaining in the period. Jujhar Khaira rounded the goal, drawing the attention of three Canadiens on the wraparound attempt while Shore was left alone on the doorstep to stuff home the rebound for his third of the season.
Price robbed Alex Chiasson from close range on an Oilers power play midway through the second period. Price did again on another Edmonton man-advantage late in the period, stopping Leon Draisaitl and then stacking both pads to deny Chiasson.
Moments later, Nurse overpowered Price with a shot from the point with 11 seconds remaining in the period and four seconds left on the power play. It was his 13th goal of the season, moving him past Arizona's Jakob Chychrun (12) for the league lead among defensemen.
NOTES: With the addition of Staal, Montreal becomes the fifth team in the modern era to dress six-plus Stanley Cup winners after having none the previous season (excluding expansion teams in their inaugural campaign). Staal, who won the Stanley Cup in 2006 with Carolina, joins Corey Perry (2007, Anaheim), Michael Frolik (2013, Chicago), Tyler Toffoli (2014, Los Angeles), Jake Allen and Joel Edmundson (both 2019, St. Louis) as new Habs who have won the Cup.
Oilers: Travel to Ottawa to take on the Senators or Wednesday and Friday.
Canadiens: Visit Toronto on Wednesday.
More AP NHL: https://apnews.com/hub/NHL and https://twitter.com/AP-Sports
Copyright 2021 STATS LLC and Associated Press. Any commercial use or distribution without the express written consent of STATS LLC and Associated Press is strictly prohibited.