Following a mostly ceremonial faceoff, the teams skirmished at center ice.
Ovechkin opened the scoring on a power play with 1:47 remaining in the first period. Jesper Fast tied it with 4:39 left in the third, tipping Kevin Hayes' shot past goalie Braden Holtby.
Holtby made 31 saves, and Lundqvist stopped 27 shots.
Game 2 is Saturday at Madison Square Garden.
The Capitals ended a two-game losing streak against New York in the playoffs. Washington also lost three of the four regular-season meetings with the Rangers.
In the first period, with New York center Dominic Moore in the penalty box for holding, Ovechkin carried the puck into the offensive zone and whipped a shot past Lundqvist from the top of the left circle for his third goal of the playoffs.
Ovechkin's goal was Washington's first playoff score against Lundqvist in 138:13 dating to Game 6 of the 2013 East quarterfinals.
In 27 playoff games against the Rangers, Ovechkin has 11 goals and 10 assists. He led the NHL during the regular season with 53 goals, 25 power-play goals and 11 game-winning goals.