Lucas Raymond scored twice, and Alex Lyon made 32 saves for the Red Wings (34-28-6), who have lost eight of nine.
Detroit won 4-1 against the Buffalo Sabres on Saturday to end a seven-game losing streak.
“I think there's a little recipe for this group,” Red Wings coach Derek Lalonde said. “We played extremely well last night. Things we've been trying to emphasize, better D-zone, managing the puck. There were moments in tonight's game that got away from us. This is a really talented team. If you're not hard on them, if you don't take away their time and space, that's what it's going to look like."
Smith put Pittsburgh ahead 1-0 at 10:10 of the first period. Marcus Pettersson shot into the slot, where it went in off Smith’s right leg.
Raymond tied it 1-1 at 15:39, scoring for the fourth straight game on a wrist shot glove side from the left face-off circle.
Crosby gave the Penguins a 2-1 lead at 18:53. Red Wings forward Andrew Copp tapped the rebound of a shot from Ryan Graves toward Crosby, who used his skate to collect it and shoot a backhand five-hole.
“Been a while,” Crosby said. “Nice to see it go in. Hopefully some more will start to go in. There are some huge games coming up. It would be nice to see some more go in the net.”
Puustinen made it 3-1 at 19:16, taking a pass from Smith below the goal line and scoring with a snap shot from the right circle.
Pittsburgh outshot Detroit 15-4 in the first.
Michael Bunting extended the lead to 4-1 at 9:34 of the second period, diving to score on a rebound at the top of the crease off a shot from Puustinen.
Christian Fischer cut it to 4-2 at 17:35 with a one-timer from the slot.
“We give their skill and their guys that much time on the power play, they get touches, they get looks, they get momentum, they get rolling," Lalonde said. "Our first power play ended up being in the second period, we get five or six shots on it. We dominated the next two shifts, and we get a goal.”