New Jersey tied the game 2-2 at the six-minute mark of the third period as Zidlicky beat Bishop with a wrist shot from the left circle.
After a scoreless overtime period, Alfredsson scored on Ottawa's second shootout attempt. The Devils' Travis Zajac tied it 1-1, then Zibanejad's goal and Bishop's save on Loktionov ended the game.
Despite being outplayed and outshot 12-4 for much of the second, the Senators managed to escape the period leading 2-1.
The Devils tied the game after Loktionov was able to find the puck during a scramble in front of Bishop, but Greening regained the lead scoring with just 17 seconds remaining in the period.
Kyle Turris split the Devils defence with a shot and Greening jumped on the rebound to beat Brodeur stick side.
Ottawa jumped out to an early 1-0 lead scoring just two minutes into the game as Phillips picked up his fifth of the season. The Senators are 52-11-4 when Phillips scores.
Zibanejad won the faceoff in the Devils end, got it to Mike Lundin who passed it to Phillips whose wrist shot beat Brodeur cleanly.
Ottawa had a number of great chances on odd-man rushes, which led to Devils coach Peter DeBoer to call a timeout midway through the period. The move paid off as the Devils settled down.
Sergei Gonchar saw his ten-game point streak come to an end Monday night.
Notes: The Senators signed forward Buddy Robinson to a three-year contract Monday. C Jim O'Brien, LW Kaspars Daugavins and D Andre Benoit were a healthy scratch for the Senators. Ottawa is without G Craig Anderson (ankle sprain, indefinite), D Erik Karlsson (Achiles, out for season), C Jason Spezza (upper body injury, indefinite), LW Milan Michalek (knee, indefinite), D Jared Cowen (hip, out for season)… The Devils are without C Dainius Zubrus (wrist surgery, 2-4 weeks), D Henrik Tallinder (lower body, injured reserve), RW Ilya Kovalchuk (sprained shoulder, indefinite), LW Alexei Ponikarovsky (lower body, indefinite). RW Krystofer Barch and D Mark Fayne were a healthy scratch for the Devils.