"This is one of those games which can turn this ship around," said Joseph, who played in his 876th NHL game. "We played a very cohesive game and put it all together."
Scatchard had a goal and an assist in the second period for his first two-point game this season. Nagy assisted on each of the Coyotes' first two goals. Derek Morris and Owen Nolan also scored for Phoenix.
The Coyotes killed off all six penalties and allowed few scoring chances, limiting New Jersey to 10 shots through the first two periods.
"I don't think we caught (the Devils) at their best," Phoenix coach Wayne Gretzky said. "But give our guys credit for a very strong effort. We made some smart decisions, and those decisions put us in good positions to make things happen. I'm thrilled by our effort and the unselfishness of our team."
Conversely, Devils coach Claude Julien saw little to be encouraged about.
"We were totally outworked," he said. "We did not look like a team in sync. In fact, we were all over the place, and it showed on the scoreboard."
John Madden scored on a goal-mouth scramble for the Devils at 1:58 of the second period.
The Coyotes broke on top just past the halfway mark of the opening period. Jeremy Roenick, on his off wing, fed a crashing Morris in the right faceoff circle with a perfect pass. Skating into the pass, Morris beat Devils goalie Martin Brodeur with a rising slap shot at 13:17.
Despite managing only one shot through the first 12 minutes, the Devils generated several good scoring chances against Joseph late in the opening period. First, he stopped Jay Pandolfo from the slot on a 2-on-1 and later gave Brian Gionta, with the puck on his stick, no room in front of the net.
"This was a weird game, and hard for us to get going," Brodeur said. "For some reason, we tried to make plays, but turned the puck over. It cost us a couple of times and we just could not recover."
After Madden tied it, Phoenix responded with consecutive goals.
First, Scatchard gave the Coyotes the lead when he tipped in Ed Jovanovski's shot from the right point at 4:30. Then, Nolan beat Brodeur short-handed when he converted Scatchard's pass off the right-wing boards at 10:38. The short-handed goal was the Coyotes' second of the season.
"We changed the way we killed penalties, and that was the key to the win," Scatchard said. "We were much more aggressive than in past games, and made them pass four or five times to try and get in a position for a decent shot. We have been playing good hockey of late, but not showing any results."
During the final period, the Devils managed only one solid chance against Joseph. Scott Gomez was stopped in close with about six minutes remaining.
Mike Ricci returned to the Phoenix lineup for the first time this season. The center was out with a neck injury and rehabbed for two games at San Antonio in the AHL. ... The one goal allowed by Joseph tied his best outing of the season. He lost 1-0 to Dallas on Nov. 9. ... The crowd of 12,883 was the third-lowest for a Phoenix home game this season, and the third crowd under 13,000 in 12 home games.