Ottawa led 3-1 with 11:29 remaining. Bozak tied the game with 1:24 to go in the third period, tipping in a James van Riemsdyk pass.
The Senators (37-25-12) have lost two in a row and are one point back of the Boston Bruins for the second wild card into the Stanley Cup Playoffs from the Eastern Conference. Boston, which has played one more game than Ottawa, defeated the New York Rangers 4-2 Saturday.
Ottawa goaltender Craig Anderson made 27 saves in his third appearance since Jan. 21; he was out with a deep bruise on his hand.
Senators goaltender Andrew "The Hamburglar" Hammond, who went 14-0-1 in his first 15 NHL games with Anderson sidelined, is out with a lower-body injury.
"I think we did a lot of good things, especially in the first two periods," Anderson said. "It's a 60-minute game and you have to play it out. You give teams time and space and it comes back to bite you. We gave their best players time and space and that's what it came down to. Bozak, Phil Kessel and James van Riemsdyk had a pretty good game and we allowed them to."
The Senators host the Florida Panthers on Sunday, and Anderson said he would be ready to play again.
"There were no issues as far as how I felt out there," Anderson said. "There's not really much I'd change as far as how I did things. We'll see how it is [Sunday] and go from there, but tonight there are no issues."
Toronto (28-42-6) ended a seven-game losing streak.
It is the first time in Bozak's six-season NHL career he has scored 20 goals; he has 22. Bozak scored 19 last season before he was injured for the final 25 games. The center on the Maple Leafs' top line, he had one goal in his past 11 games.
"Obviously the way it has been going it was nice to get a few bounces," Bozak said. "We feel like we haven't been getting a lot of bounces for a while, but that is no excuse because we haven't been playing well enough either. It was nice to get a bounce, and then I felt like things started clicking for us. Hopefully we can carry on like that for the rest of the year and finish strong as a team."
Brewer's game-winner came on a scramble in front of the net. It was his first goal in 12 games with the Maple Leafs after being acquired from the Anaheim Ducks prior to the NHL Trade Deadline.
"It was a hack-fest out there with the puck bouncing," Brewer said. "I just swatted at it and it went in."
Toronto is 3-10-1 in March.
"It has been a grind," Brewer said. "I'd be lying if I said otherwise. I think we're starting to play a little bit better even though there's still plenty of work to do. We stuck with it tonight, which was nice."
Anderson made his mark stopping Toronto's Joakim Lindstrom on a penalty shot at 11:14 of the first period. After Lindstrom was cut down driving to the net, he tried to deke to his forehand, but Anderson made the save look easy.
The Senators took the lead with two goals in 1:34 of the second period. Mark Stone scored his 19th of the season at 4:25, beating Toronto goalie James Reimer with a wrist shot to the stick side after taking a drop pass in the slot from former Maple Leafs forward Clarke MacArthur.
Curtis Lazar made it 2-0 at 5:59 with a shot to Reimer's stick side from the slot.
Bozak scored his 20th of the season at 4:13 of the third period. Van Riemsdyk drove hard to the net and spotted Bozak in front of the net. Bozak escaped the checking of Senators defenseman Mark Borowiecki and tapped the puck past Anderson.
With MacArthur off for boarding, Bozak scored his second from the side of the net to make it 3-2 at 11:29.