Chris Kreider scored, and Georgiev made 22 saves for the Rangers.
"Two different games," said Gallant, who was hired June 16, replacing David Quinn. "I thought the first 33 minutes we were really good even though were behind 2-0, a couple of power play goals that they scored. They got the bounces on both of them. But I thought we played a [heck] of a 33 minutes and after that we weren't too good."
The Capitals were 3-for-6 on the power play. The Rangers were 1-for-5.
"Especially against a team like Washington, you don't want to take too many penalties," New York center Mika Zibanejad said. "That's something we have to get better at in the future and think about, but without taking anything away from our aggressiveness and playing hard."
Oshie gave the Capitals a 1-0 lead at 4:39 of the first period when he reached behind Georgiev and swept in a loose puck during a power play.
Capitals defenseman Nick Jensen scored at 9:22, but the goal was overturned after the Rangers challenged for offside.
Schultz made it 2-0 with a power-play goal from the point at 12:38 of the second period.
Washington scored 24 seconds later during a 2-on-1 after Oshie sent a drop pass between his legs across to Lapierre, who scored from the left face-off circle to increase the lead to 3-0 at 13:02.
"I knew when I called [for] the puck he was going to pass it to me, and I got lucky. I just put the puck at the net," Lapierre said. "It was just so fun. The crowd was so incredible, and for a first game I don't think I could have asked for a better one, and we got a great win."