The Maple Leafs (26-20-11) had three power plays in the first period, including a two-man advantage for 22 seconds, but could not score.
The Senators, who led 2-0 in the first period, tied it 3-3 at 5:32 of the third when Hoffman's wrist shot from the high slot fooled Andersen.
Ottawa took a 4-3 lead at 6:26 on a power play when Brassard tapped in a rebound during a scramble in the goal crease. Stone scored into an empty net with 1:50 left to make it 5-3, and Brassard scored his second to make it 6-3 with 45 seconds to go.
"We had ample, ample opportunity to win the game down the stretch," Maple Leafs coach Mike Babcock said. "We had lots of push at the net and we didn't score and they did. In the end, when you look at the score, 6-3, you don't think it was a very good hockey game, but I think it was a pretty good hockey game. I think we can play way better though. I think we can play better structurally defensively, so we have to get better.
Ottawa built a 2-0 lead on goals from Chris Wideman and Ryan Dzingel 20 seconds apart beginning at 17:26 of the first period.
Toronto began its comeback with two goals in 3:14 of the second period. Morgan Rielly made it 2-1 when he scored on a wrist shot from the blue line on its 22nd shot of the game at 14:38. Nazem Kadri tied it 2-2 at 17:52.
William Nylander gave Toronto a 3-2 lead at 2:04 of the third period.
"We were expecting a push after the first period and we got one," Brassard said. "We stuck with it and showed a lot of character. We've been doing that all year long. Our goaltending was good, we had some key saves at key moments of the game."