Mika Zibanejad scored two goals, Filip Chytil and Chris Kreider each had a goal and an assist, and Georgiev made 33 saves for the Rangers (6-2-3).
"I think we did a good job getting the lead, we had a three-goal lead, and I think we did a good job up to that point," Zibanejad said. "I don't know if we get too passive and try to protect the lead in a little too much of a defensive way, but that one obviously stings. They have some good players that can make plays, but I think we just have to find a way to battle through that and end it there."
Chytil scored at 1:53 of the first period to give the Rangers a 1-0 lead.
Zach Hyman tied it 1-1 at 18:11, getting his own rebound off a point shot from Barrie.
Kevin Rooney gave New York a 2-1 lead at 2:52 of the second period following turnover inside Edmonton's zone by Warren Foegele.
Kreider pushed the lead to 3-1 at 4:11 with a shot from the right circle on a rush, and Zibanejad scored on the power play at 6:54 to make it 4-1.
"I'm disappointed no doubt, the way it happened. We played a great 35 minutes, then we sort of started watching them a little bit," Rangers coach Gerard Gallant said. "We were a little too passive. When you give teams time and space, they're going to make their plays, especially with their talent. We played too much in the D-zone the last 25 minutes of the game, and it caught up to us."
Puljujarvi cut it to 4-2 at 14:41 of the second, taking a stretch pass from Ryan Nugent-Hopkins and scoring with a slap shot from the right circle on a power play.
"That gave them some good momentum," Rangers defenseman Jacob Trouba said. "We had some good stretches on the PK, but giving up two on the PK, it's not a good night. We knew their PP was powerful coming in, and we wanted to do better than we did for sure."