Brett Connolly and Evgeny Kuznetsov scored for the Capitals (10-9-1), who gave up 12 goals and lost both games on their road trip. Philipp Grubauer had 22 saves.
"I just don't like the last two games," Capitals coach Barry Trotz said. "For whatever reason, we've had poor starts. We haven't been ready out of the gates. We haven't worked hard in our battles."
Landeskog scored 17 seconds into the first period, beat Grubauer on a penalty shot at 4:59 of the third period, and got the hat trick on the power play with 25.2 seconds remaining. He has eight goals this season.
"It felt good to get that third one," Landeskog said. "I had a couple of good looks before. It was a great team win."
Landeskog was awarded the penalty shot after Capitals defenseman Brooks Orpik slashed him on a breakaway.
"I was talking to Mikko and Nate, I hadn't really made up my mind," Landeskog said of his strategy. "Mikko said come in and shoot low blocker. I came in and I didn't really feel that, so I just slowed down and tried to flick it high blocker, and it worked. I'm not really a shootout specialist by any means, but I was able to sneak that one in."