Moulson tipped in a shot from the point by Christian Ehrhoff for his 12th goal of the season.
The Sabres went on the power play when Jets rookie forward Eric O'Dell, making his NHL debut, was called for hooking.
"He took a penalty and it wound up in our net," Jets coach Claude Noel said. "How do you analyze that? It's what happens sometimes."
The Sabres had scored one goal in seven of their past 12 games. They've scored four goals in a game one other time this season; a 5-4 shootout win against the San Jose Sharks on Nov. 5.
"I think it was due, definitely," Ehrhoff said. "We only scored one goal, I don't know how many games in a row. We had a lot of chances before. Today we just got a few at a good time."
"We got outworked," Jets forward Olli Jokinen said. "They got that confidence in the first period. They probably felt they could be up three or four goals. Without our goalie, that would've been the case after the first."
Scheifele scored his second of the game and sixth of the season 36 seconds later to give the Jets a 2-0 lead. He stole the puck at center ice from Sabres forward Matt D'Agostini, skated in and sent a wrist shot past Miller.
"In the second period we managed the puck way better," Noel said. "We forechecked way better. Scheifele made a couple of really nice plays. He was probably our best player."
The Sabres made it 2-1 when captain Steve Ott scored his fourth of the season at 10:16 of the second. The play started when defenseman Mike Weber's pass from the blue line went to forward Drew Stafford; his shot bounced off Jets goalie Ondrej Pavelec to Ott, who scored on the rebound.
Ott's goal broke his 12-game streak without a point, dating to a goal and an assist against the Toronto Maple Leafs on Nov. 15.
"I think that lifted us up again and it kind of snowballed from there," Ehrhoff said.
"Tonight the way he played, he the way he skated, the way he went to the net, the way he used his body," Sabres interim coach Ted Nolan said of Foligno. "I think one shift he had three good hits. … He plays that way, he's going to be a powerful force in this League."
Sabres forward Matt Ellis made it 4-2 at 13:38 with his first of the season. Ellis batted a rebound of a Kevin Porter shot out of midair and past Pavelec. Ellis was called up from Rochester of the American Hockey League on Dec. 14, and Porter was recalled from the minors before the game. The goal was Ellis' first in the NHL since Jan. 13, 2012.
"My linemates made it easy for me tonight, but they did a good job getting pucks down low and getting it to the net," Ellis said. "I was a recipient of going to the net, and good things happen, and I was able to find the back of the net. It was great getting to contribute to a victory."
Ellis was on a line with Porter and another former Rochester teammate, Brian Flynn.
"That so-called fourth line, they're one of four for us," Nolan said. "One thing we want to do is try to create a balanced attack. Tonight with Flynn, Ellis and Porter, they did a great job."
Miller made 25 saves; Pavelec had 27. The loss was the Jets' fourth in regulation against the Eastern Conference. They're 9-4-1 against Eastern teams and 15-16-5 overall in their first season as a member of the Western Conference.
"You've got to be able to close the games," Jokinen said. "At the end of the day as a player you've got to control the things you can control. You can't worry about other guys or anything else. You've got to control your own emotions, your own focus. That's the way the game's always been played and that's the way it's going to be played for a long, long time. We're all going to be gone and you've got to play it the right way and you get bounces and you have to be willing to do that every shift."
The Sabres (8-23-3) host the Boston Bruins on Thursday. The Jets head home to face the Florida Panthers on Friday.