Vanek scored on a backhander against Florida's Roberto Luongo in the top of the second round. After Aleksander Barkov evened the shootout at 1-1, Nielsen slipped a puck through Luongo, actually making contact with it twice.
Following a short review, the call was ruled inconclusive, giving the Red Wings a 2-1 lead. Jared Coreau, making just his second NHL start, then stopped Nick Bjugstad to give Detroit the shootout win.
"Yeah, (the goal) was a little lucky," Nielsen said. "It wasn't my plan. I was going to get it all the way back on my forehand.
"I think he was biting on the backhand and I thought I had the net, but it rolled off my blade but luckily it went through his five-hole."
Coreau was called up from Grand Rapids of the AHL when Jimmy Howard injured his knee on Tuesday.
"It was a blur. They have some great shootout players," said Coreau, who made 31 saves in his first win.
"I got lucky on the first one -- post, post (on Vincent Trocheck). Honestly, I didn't feel that great on the third one, I just got a piece of it."
Florida also had a goal by Keith Yandle overturned when it was ruled that Bjugstad, who was in the crease, failed to give Coreau room to work to make the save.
"By the rulebook, they were right," Florida coach Tom Rowe said.
Florida jumped in front on its third power play of the first period. With Detroit's Xavier Ouellet in the box for hooking, Smith fired a shot from above the left circle that bounced off Danny DeKeyser's stick with Florida's Trocheck screening Coreau.
The puck tipped off the edge of Coreau's glove for Smith's seventh goal at 17:40.
The teams exchanged goals in the first 89 seconds of the second.
Mantha tied it just 12 seconds into the period when he grabbed a loose puck and made a move to beat Luongo low on the stick side for his sixth goal. Demers then blasted a long shot from near the blue line that beat Coreau high over his right shoulder for the defenseman's sixth goal at 1:29.
Smith, a 25-goal scorer last year in his first season with the Panthers, fell to the ice and clutched his face after taking a hard hit in the defensive zone from Niklas Kronwall 2:51 into the period.
He stayed down on the ice in pain and had blood from a cut on the inside of his visor when he skated off into the dressing room. Rowe said he would be out with a concussion for about a week.
MacKenzie one-timed a cross-ice pass from Matheson at 10:27 for his fourth goal, giving Florida a 3-1 advantage.
"It's frustrating. It was a weird game in that momentum seemed to go back and forth a few times," MacKenzie said. "We can talk about breaks, but we've got to make our own. We made some mistakes and it cost us."
Miller cut the deficit to 3-2 with his third goal at 1:36 of the third on a sharp-angle shot from the left circle. Detroit completed the rally when Nielsen beat Luongo in close for his eighth goal at 13:51.
Detroit general manager Ken Holland announced Howard will miss four to six weeks. Howard is 5-7-1 with a 1.96 goals-against average and a .934 save percentage. ... The Red Wings scratched Ds Mike Green and Brian Lashoff, and RW Tomas Jurco. ... Florida sent G Mike McKenna back to Springfield of the AHL after calling him up Thursday on an emergency basis. ... The Panthers scratched D Dylan McIlrath, C Denis Malgin and LW Shawn Thornton. ... Philadelphia Union and U.S. men's national team player Alejandro Bedoya was in attendance. The midfielder/winger graduated from nearby St. Thomas Aquinas High School in Fort Lauderdale.
Red Wings: Host Buffalo on Tuesday night.
Panthers: Host Toronto on Wednesday night.