The victory not also ended Washington's four-game losing streak, but it was also the 200th of Boudreau's NHL coaching career. He took over the Capitals in November 2007, promoted from AHL affiliate Hershey when Washington fired Glen Hanson on Thanksgiving Day.
"I would have traded 199 of them for that one today," Boudreau said. "When you don't win for any length of time, it gets tougher. So it came at a good time."
The Capitals entered Monday in a serious slump, having lost six of their previous seven games, including a 7-1 setback at Toronto on Saturday. But Backstrom broke a 2-2 tie only 1:52 into Monday's final period, lifting the puck past goalie Jason LaBarbera from near the net.
Joel Ward and Alex Ovechkin were credited with assists -- Ovechkin's first point in five games. That ended only the second four-game point drought of Ovechkin's NHL career; the other came in February 2007, and he's never had a longer stretch.
Backstrom earned an assist on Laich's goal at the 7:06 mark of the third period during a 5-on-3. That was Washington's first power-play goal in six games and made it 4-2.
"It's always fun to be back on track," Backstrom said. "It makes it much easier if you win hockey games to come to practice the next morning and smile a little bit more."
Lauri Korpikoski's second goal of the game, with just under nine minutes left, cut the Coyotes' deficit, but they couldn't manage to pull even.
"We kind of shot ourselves in the foot," Phoenix coach Dave Tippett said.
Phoenix's first two goals came during power plays for Washington. Radim Vrbata's came on a classic shorthanded breakaway, while Korpikoski's first of the night came via penalty shot after he was interfered with during a shorthanded breakaway of his own.
The Capitals were coming off a 0-3 road trip, but improved to 7-1-1 at home this season.
Vrbata put the Coyotes ahead with his team-high ninth goal of the season 4:33 into the game. Carlson was at the point on the power play when he fumbled away the puck to Vrbata, who beat Vokoun 1-on-1 with the Coyotes' first shot of the evening.
In the final minute of the first period, Washington's Matt Hendricks got into a fight with Kyle Chipchura, bloodying the Phoenix forward's face. The clash drew loud cheers from the stands when it was shown on the above-ice video board, and if it was an attempt by Hendricks to fire up his teammates, it didn't pay immediate dividends.
Instead, the Coyotes grabbed the next goal, going ahead 2-0 on Korpikoski's penalty shot 7 1/2 minutes into the second period. The penalty was awarded when Capitals defenseman Dennis Wideman hooked Korpikoski, who had broken alone into Washington's zone.
Suddenly, though, the Capitals came alive. One second after that power play elapsed, Carlson put in a slap shot from near the blue line, an unassisted goal less than nine minutes into the period that made it 2-1. And about three minutes later, Eakin put the puck past LaBarbera to tie the game.
"I think any kind of win when you've lost four games in a row is what you want," Boudreau answered. "So we'll take anything."
Material from wire services and team media was used in this report.