Legace stopped 15 shots, but left on a stretcher late in the second period after colliding with a pair of players. Hasek stepped in and made four saves over the final 25 minutes.
Legace was involved in a collision with teammate Fredrik Olausson and Nashville defenseman Kimmo Timonen, who tangled up in front of the net, lost their balance and fell on the goalie's head.
Defenseman Steve Duchesne scored the game's only goal at 2:35 of the second period. Sergei Fedorov sent a pass from behind the net to the slot, from where Duchesne put a wrist shot between Mike Dunham's pads.
Dunham recorded 32 saves, including 10 in the final period, to keep the Predators close. He stopped Fedorov on a breakaway with 14 minutes to play and fended off a goalmouth scramble at 2:36 of the third period.
Dunham also turned away a point-blank shot by Steve Yzerman with 40 seconds to play.
But Hasek was equally as good in the third period, with the help of the Red Wings' defense, and preserved the team's first shutout of the season.
Martin Erat had a chance to tie the game with 1:46 left. Erat pulled Hasek out of the net as he attempted to score on a wraparound, but defenseman Nicklas Lidstrom slid along the right side of the net to block the shot.
The Red Wings improved to an NHL-best 9-2-0 this season -- 4-0 on the road.