"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.