"It was just two outstanding efforts by David, creating the goal by (Chris) Gratton and the goal that he scored himself," Coyotes coach Bob Francis said.
Tyson Nash also scored for the Coyotes and former Devils goalie Sean Burke made 27 saves, including 14 in the first period. Phoenix won for the second time in six games (2-3-1).
Jay Pandolfo scored for the Devils, whose winless streak was stretched to four (0-2-2). New Jersey has a tie and a loss since a 13-game unbeaten streak that tied the team record.
"It's only a couple of loses in what, 16 games?" Devils captain Scott Stevens said. "I don't know if we're in a slump yet."
With the game tied at 1 in the third period and the teams skating four-on-four, Tanabe fed Gratton for the go-ahead goal at 11:44.
Tanabe added an insurance goal 66 seconds later as he ripped a shot from the left faceoff dot over the right shoulder of goalie Martin Brodeur to cap a pair of remarkable shifts.
"After the second one, all I could do was come to the bench and smile," Tanabe said. "They were a couple of pretty special shifts, especially against a team that didn't allow a lot of opportunities all night long."
The Devils, returning home from a four-game road trip, needed half a period to get untracked. Their first shot came 9:45 into the contest and triggered the 14-shot outburst for the remainder of the period.
Burke was sharp, making point-blank stops on Turner Stevenson, Grant Marshall, Brian Gionta and Pandolfo to keep the game scoreless after one period.
"I was trying to keep my team in the game," Burke said. "They have such a good team, I wanted to make sure I had a strong first period."
The teams exchanged goals in the second.
Phoenix snapped the deadlock at 3:25 on an acrobatic goal by Nash. Cale Hulse fired a long-range shot that Brodeur blocked. The loose puck landed in the crease, and Nash dived over Brodeur to knock it into the net.
New Jersey pulled even at 9:02 as Pandolfo batted Tommy Albelin's point shot past Burke.