Buffalo got a score from Jason Pominville in the second round, leaving the Senators needing a goal to open the third round to keep the game going.
Daniel Alfredsson scored for Ottawa but Ennis answered to end the contest and give Buffalo its eighth victory in 12 games. Two of the Sabres' four losses in that stretch came in overtime.
Buffalo remained four points behind Washington in the fight for the Eastern Conference's final playoff berth.
Ennis scored in the second period and Buffalo came up with goals from Marcus Foligno and Nathan Gerbe.
Ryan Miller stopped 33 shots in the win.
With the one point they earned Saturday, the Senators moved within two of Boston at the top of the Northeast Division.
Ben Bishop made 38 saves for Ottawa, which had won two in a row.