Antoine Vermette was awarded a goal in the final minute after Boston's Marc Savard threw his stick while the Ottawa forward had a breakaway on an empty net.
Glen Murray had the lone goal for the Bruins (22-21-4), who were outshot 36-30.
The 19,846 in attendance at Scotiabank Place sat through two periods before a goal was scored and then saw the red light turned on four times in the final period.
Eaves opened the scoring at 2:17 to give the Senators (29-20-2) the lead. After Jason Spezza tipped the puck into the slot, Eaves spun and fired a low shot that went through the legs of Bruins goalie Tim Thomas.
Spezza was back in the lineup for the first time after missing Ottawa's last 14 games with a knee injury, but as expected, he wasn't part of the top line with Dany Heatley and Daniel Alfredsson. Those two played between Chris Kelly, who was solid in Spezza's absence. Spezza was part of a line with Eaves and Vermette, who also assisted on the first goal of the game.
A little less than six minutes after Ottawa took the lead, Neil made it 2-0 when he converted a pass from Peter Schaefer and beat Thomas to the glove side.
Boston then cut it to 2-1, when Murray beat Ray Emery at 8:44 on a setup from ex-Senator Zdeno Chara. It was the first goal allowed by the Senators and Emery in more than 157 minutes.
Notes: With the win the Senators have now won 12 of their past 15 games and collected points in 13 of those games . . . Ottawa visits Montreal on Monday night . . . Boston has dropped nine of its past 12 game and host the New York Islanders on Monday.