Senators coach Paul MacLean didn't excuse his team's poor performance for much of the game.
"We played way too much of the game in our end," MacLean said. "I thought they were way better than we were for the majority of the game.
"At the end of the day we did enough to win the game."
Devils coach Peter DeBoer would have liked to have seen his team's effort rewarded with two points, but had no problem with the team coming out on the losing end of the shootout.
"It was a gutsy effort by us," DeBoer said. "I don't look at any points as disappointing. It's a tough building. I thought our guys played their hearts out and deserved better, but we'll take the point and keep moving forward here."
New Jersey tied the game 2-2 at 6:05 of the third period as Zidlicky beat Bishop with a wrist shot from the left circle.
After overtime, 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 beat the Devils defense with a shot and Greening jumped on the rebound to beat Brodeur stick side.
"Tonight wasn't our best hockey by far," Bishop said. "I don't think we played our best, the last five minutes especially. We turned too many pucks over."
Ottawa jumped to an early 1-0 lead scoring just two minutes into the game as Phillips picked up his fifth of the season.
"It's one of those streaks, records that I can't really explain why," Phillips said. "I can't explain why it happens or why it's so favorable, but it's been fun."
Zibanejad won the faceoff in the Devils end, got it to Mike Lundin who passed it to Phillips whose wrist shot beat Brodeur.
Senators defenseman Sergei Gonchar had his 10-game point streak come to an end Monday.