Tavares tied it 4-4 at 17:22 of the second.
"The players decided it was important," Toronto coach Sheldon Keefe said. "That's it. We know how to play. When we play, we show what we are capable of."
After allowing three goals on 14 shots in the second, Reimer stopped all 18 shots faced in the third period.
"That third period by him was unbelievable," Larkin said. "We were happy to get it to the intermission with how the second period went. Our message was, 'Win a period, win a game.' It ended up going a little extra but we got two points and Reimer was a huge part of it and in his 500th game, I can't be happier for him."
Marner gave the Maple Leafs a 1-0 at 8:35 of the first period on the power play.
DeBrincat tied it 1-1 at 10:02 when he shot blocker-side on a 2-on-1 with Lucas Raymond.
Simon Edvinsson gave the Red Wings a 2-1 lead at 13:39 when he took a pass on a give-and-go with Perron and scored from the left hash marks.
DeBrincat made it 3-1 at 17:37 on the power play when he put in the rebound of J.T. Compher's shot.
Perron extended the lead to 4-1 at 19:41 when he took a pass from Kane, who was behind the net, and scored from the right face-off dot.
"We were obviously playing a very desperate team, a team that's fighting for their lives and they were just much more desperate than us in the first and I just thought we got playing in the second, took it one shift at a time and we were able to battle back," Matthews said.
NOTES: Larkin scored his ninth overtime goal, tying Steve Yzerman and Brendan Shanahan for the second most in Red Wings history behind Sergei Fedorov (12). ... Matthews has 107 points this season, setting a new career high. He had 106 points in 2021-22 (60 goals, 46 assists in 73 games). Matthews is the sixth different Toronto player to have an eight-game goal streak. ... Toronto forward Bobby McMann left the game after the first period with a lower-body injury and did not return. There was no update after the game. Defenseman Jake McCabe took a puck to the face at 16:46 of the second period and did not return. Keefe said he was fine but didn't return due to swelling.