Anderson drew Montreal within 2-1 at 1:45 of the third when he deflected Corey Perry's pass.
Tatar tied it 2-2 with a wrist shot past Smith's glove from the right face-off circle at 5:23.
"We weren't on our toes as much as we should," Oilers coach Dave Tippett said. "Then once they got the first one, we got back to playing a little bit, and a mistake cost us the second one. So the best way to protect the lead is to play in the other team's end, and the first five minutes we were passive, didn't didn't do enough forechecking to give us the momentum we needed."
Nurse put Edmonton up 2-0 when he scored on the power play with 11 seconds remaining in the second period. It was his 13th goal, the most by a defenseman in the NHL this season.
Smith made 14 saves in the first period, and Devin Shore scored on Edmonton's fifth shot on goal to make it 1-0 with 21 seconds remaining. He was left unchecked on the right side and scored into an open right side after Jujhar Khaira carried the puck behind the net before setting him up with a backhand pass across the crease.
NOTES: Staal is the first player in Canadiens history to score in overtime in his debut with them. He is one of 10 NHL players to do so in his first game with a team since 1983-84, when regular-season OT was reintroduced, and the third this season (Kirill Kaprizov, Minnesota Wild, Jan. 14; Aleksi Heponiemi, Florida Panthers, Jan. 30). ... Montreal forward Brendan Gallagher left the game after he was struck by teammate Alexander Romanov's shot in the first period. Coach Dominique Ducharme said he has a broken thumb. ... Edmonton will complete a three-game road trip with a two-game set at the Ottawa Senators beginning Wednesday. The second game was moved to Thursday from Friday by the NHL on Monday. … With Staal, the Canadiens became the fifth NHL team since 1943-44 to have at least six players who previously won the Stanley Cup after having none the previous season. Staal won the Cup with the Carolina Hurricanes in 2006. He joined Perry (Anaheim Ducks, 2007), Michael Frolik (Chicago Blackhawks, 2013), Toffoli (Los Angeles Kings, 2014) and Jake Allen and Joel Edmundson (St. Louis Blues, 2019). The 1994-95 Blues, with seven Cup winners, are the only team that did not win the Stanley Cup the previous season. The others are: 1974-75 Philadelphia Flyers (19 Cup winners), 1980-81 New York Islanders (22) and 1989-90 Calgary Flames (20).