They rallied for a 2-1 win in double overtime to cut their NHL Eastern Conference semi-final series deficit to 2-1.
Tyler Kennedy's goal with 1:07 left in the second period looked to have been enough to hand the visitors victory at Scotiabank Place in game three, which would have put them on the brink of a finals place.
But Senators captain Daniel Alfredsson had other ideas and tied the game with a short-handed goal 28.6 seconds from the end of regulation, before Colin Greening netted the winner at 7:39 of the second overtime.
The best-of-seven series resumes with game four in Ottawa on Wednesday.
Also on Sunday, the Boston Bruins were twice pegged back before surging clear for a 5-2 triumph to go 2-0 up in their EC semi-final series against the New York Rangers.
Torey Krug and Gregory Campbell saw strikes cancelled out by Ryan Callahan and Rick Nash, but Johnny Boychuk restored Boston's advantage at 12:08 in the second period and third-period goals from Brad Marchand and Milan Lucic sealed victory at TD Garden.
Game three will be played at Madison Square Garden on Tuesday.