Ovechkin put the Capitals ahead 1-0 at 8:55 of the second period, beating Vejmelka blocker side with a one-timer from the left circle off a pass from Trevor van Riemsdyk on a power play.
"Obviously, it's a huge moment. The last couple games it was big milestones and, unfortunately, we didn't get the win," Ovechkin said. "But it's a special moment, and it's nice to have both my kids, my wife, all the fans (here). My parents watched in Moscow."
Mantha made it 2-0 at 3:23 of the third period with a spinning backhand in front after Gostisbehere's clearing attempt ricocheted to him off the skate of Dysin Mayo.
Brown pulled the Coyotes to within 2-1 at 6:33 when he scored on a rebound in front.
"It was a simple play, but it's something we don't do enough," Tourigny said. "Just putting pucks at the net and throwing bodies at the net. [Gostisbehere] put the puck at the net, and 'Brownie' found the rebound."