"We didn't give them too much," Hasek said. "They had some chances and hit the post."
The Red Wings snapped a four-game losing streak.
"Been a long time around here since we've been able to talk about a win," Detroit coach Mike Babcock said.
Defenseman Mathieu Schneider praised Hasek.
"He just does what he always does," Schneider said. "He always gives us a chance to win the game."
Jean-Sebastien Giguere made 27 saves for Anaheim.
"Exactly like a playoff game," Ducks star Teemu Selanne said. "We stopped skating in the second period."
The Red Wings have already clinched a spot in the playoffs for the 16th consecutive season. That's the longest playoff streak in North American professional sports. The New York Yankees are second with a 12-season playoff streak.
"You get to this point in the season and you're just dying for the playoff to start," Schneider said. "Anytime you come up against a team that's' a potential rival in the playoffs, you want to raise your game and I thought we did that tonight."
Datsyuk scored 4:30 into the second period, putting in a rebound of Schneider's shot from the point off a faceoff. The puck also bounced off Detroit's Tomas Holmstrom, who was in front of Giguere, before Datsyuk scored.
"He's just a flat-out good player," Babcock about Datsyuk. "Whether you're killing a penalty or protecting a lead and he's dangerous offensively."
The Ducks had a breakdown on the faceoff
"I don't know what happened," said Anaheim's Corey Perry. "We made a mistake on the face-off and, before you know it, the puck is in the net."
- Detroit defenseman Chris Chelios was honored by USA Hockey before the game for playing more NHL games (1,542) than any other U.S.-born player.
- Five of Anaheim's six remaining games are on the road.
- Red Wings defenseman Brett Lebda missed the game because of a concussion.