The Sabres (14-27-3) retired Hasek's No. 39 in a pregame ceremony. Hasek won the Hart Trophy twice and Vezina Trophy six times when he played for the Sabres from 1992-2001. He also led Buffalo to the 1999 Stanley Cup Final against the Dallas Stars.
"It's nice to meet the legend and the goalie who did so much stuff for our country and for himself, so I think it's great," said Mrazek, who made 25 saves to earn the win.
Mrazek kept a game puck with a special emblem recognizing Hasek's number was retired.
"It's nice to have these memories," Mrazek said. "I have every puck from junior championships or from the Calder Cup Final. So every puck I can, it's nice to keep."
The Red Wings (23-11-9) took a 2-0 lead on goals from Helm and Nyquist.
Helm made it 1-0 with a shorthanded goal 9:42 into the first period. He picked up a loose puck and skated in on a 2-on-1 with Joakim Andersson against Sabres defenseman Andrej Meszaros. Helm held on to the puck and took a wrist shot that beat Neuvirth for his eighth goal.
"It was a great shorthanded goal by [Helm]; he really improved his angle on the 2-on-1," Red Wings coach Mike Babcock said. "We scored a nice power-play goal too, so our specialty teams were good tonight."
It was the first time in the past eight games the Red Wings scored the first goal.
"We talked about that before the game, how the last eight or so games we've been giving up the first goal," Quincey said. "So that was important for us to get a good start, get the first one and definitely after that ceremony get the legs going."
The Sabres have allowed seven shorthanded goals, the most in the NHL.
"The power play should give you some momentum," Sabres coach Ted Nolan said. "Unfortunately, the way we are, it's taking momentum away. We keep doing the same type of thing; it's not something different."
Buffalo has lost eight consecutive games in regulation, the most in Sabres history.
"We're working hard, we're just not working smart," Sabres forward Tyler Ennis. "We're beating ourselves with bad decision-making."
Nyquist made it 2-0 with a power-play goal 7:34 into the second period, 17 seconds after Sabres defenseman Rasmus Ristolainen was assessed a double minor for high sticking on Riley Sheahan. Nyquist scored his 16th goal on a rebound from a Justin Abdelkader deflection of Pavel Datsyuk's shot.
"I think specialty teams in this kind of hockey game are really important," Tatar said. "I'm glad guys did a really good job on the [penalty kill], and we tried to score some goals, we could score more on the power play, but we were happy for one."
Weber's goal was his first since April 6 against the Philadelphia Flyers, a span of 39 games.
"That's the word we used the last couple weeks is having some urgency in our game," Nolan said. "This league, there's a lot of guys given opportunities this year. When you have an opportunity, you have to do something. I always tell my boys, 'When you play, you've got to do something to let people know that you're playing.'"
Tatar made it 3-1 with 4:51 left in the second period when he put a rebound of a Quincey shot past Neuvirth for his 17th goal. Tatar was untouched in front of the net between Meszaros and Weber and had a clear shot at the puck.
"[Tatar] turned the puck over too, so it was good to see him get it back," Babcock said. "That was an important goal for us."