Varlamov protected the shutout in the closing minutes, diving to stop Markus Naslund.
After being held to three goals in two games, one at even strength, the Capitals showed how they ranked third in the NHL in scoring during the regular season.
Ovechkin cut down on his shooting after recording 19 shots in the first two games and having another 17 attempts blocked, and showed off his passing skills. He earned assists on Alexander Semin's two goals in the first period and was on the ice when Brooks Laich added a power-play tally in the second.
Former Rangers defenseman Tom Poti also scored a power-play goal with 1:25 left.
Ovechkin was everywhere, even chasing down Lauri Korpikoski on a short-handed breakaway. Ovechkin dived from behind and deftly knocked the puck off Korpikoski's stick as the Rangers forward crashed into the net.
It was a fine ending to Ovechkin's day after he was shooed away from the Capitals bench in the morning when he took a seat to watch the Rangers practice. He joked that New York was afraid of him, and then gave the Rangers every reason to be.
The NHL's leading goal scorer took the Rangers apart with the pass. He earned the second assist on Semin's first goal and made the final pass in front to set up Semin's second of the night and third straight for the Capitals in the series.
New York coach John Tortorella cautioned that the Rangers won the first two games despite not having the puck for the majority of the time. That trend continued in the opening period, and they paid for it.
Washington controlled play throughout the first, outshooting the Rangers 7-2, and used a pair of passing plays to grab its initial multigoal lead of the series.
Nicklas Backstrom, who also assisted on Semin's two goals, found his linemate with a cross-ice pass into the lower right circle for a shot that made it 1-0 at 6:57. Semin snapped the Capitals' scoreless drought at 85 minutes, 15 seconds -- dating to his early third-period goal in Game 1.
The Rangers nearly tied it when Markus Naslund's shot from the right circle was stopped by Varlamov, and the rebound came right to Ryan Callahan on the other side, but his drive at the open side kicked off the left post with 8:38 left.
Washington raced the puck the other way, as the crowd groaned at the missed chance to tie. Backstrom sent a pass from the left-wing boards to Ovechkin at the goal line. He spotted Semin in front for a quick redirect that made it 2-0 just 14 seconds after Callahan's miss.
Lundqvist made 36 saves did all he could to keep the Caps at bay, kicking out his legs, sprawling on his stomach and back and diving post to post. His best stop came when Tomas Fleischmann got free in front, but was denied on a mini breakaway by Lundqvist's quick right pad as the goalie fell forward with 12:21 left in the second.
Washington pushed the lead to 3-0 with 8:31 remaining in the second when Laich cleaned up a loose puck at the right post after Semin bounced a shot off Lundqvist's pads.
Notes: The Capitals had lost Game 3 the four previous times they trailed 2-0. They are 11-19 overall in Game 3s. ... New York also dropped Game 3 at home last year in the first round after winning the first two at New Jersey. The Rangers took that series in five games.