Just 25 seconds after Fabbri's third goal, Dallas forward Roope Hintz spoiled Detroit's shutout as he tipped a shot from Gurianov that beat Greiss to make it a 3-1 game with 10:30 remaining in the game.
The Stars made it a one-goal game with 1:20 to go when Jason Dickinson deflected a point shot from Jason Robertson for a six-on-five goal that brought the game to 3-2.
But the Wings stood tall in the final 80 seconds of the game, handling a barrage of shots from the Stars to secure the win.
Blashill said he was happy with the way his team responded to Bernier's injury and helped Greiss get comfortable.
"Our guys like Greiss a lot. He works hard, he's a great pro, he's a great person and I think it's a tough situation to go into," the Wings' coach said. "So I think our guys wanted to get him through that game. Ultimately, I think he ended up recording the win. He's a guy that, again, our guys like a lot and I think he came in a did a great job."
The Red Wings' win snapped a three-game losing streak to the Stars and gave Detroit seven points in its last five games.
Fabbri said the Wings are gaining confidence with every victory and said the team chemistry is brewing into something that could be special.
"It's going really well right now. Everyone's having fun. When you're winning, it's fun," Fabbri said. "That brings a lot of confidence in everyone's game and when you're feeding off guys, each and every line has a good shift and you want to follow it up. It's really nice that everything's been coming together. There's still a long road to go here but we're definitely on the right track."
The Red Wings will look for their third consecutive win on Saturday night as they wrap up their six-game homestand. Puck drop for the series finale against the Stars is set for 7:00 p.m. on FOX Sports Detroit and 97.1 The Ticket.