Nyquist scored 17 seconds into the game, then added the eventual game-winner on a breakaway in the third period of a 4-3 victory against the Carolina Hurricanes.
"We'll see what happens here, but hopefully that's the case," Nyquist said of remaining with the Red Wings. "Obviously, it's here I want to play, so hopefully we can work that out."
It was the first win for the Red Wings at Joe Louis Arena since Oct. 15, ending an 0-2-6 run at home and an 0-2-5 stretch overall. They were facing the right opponent: Carolina has not won at Detroit since Nov. 14, 1989, when the franchise was known as the Hartford Whalers.
"Obviously we needed this win in front of our home crowd," Nyquist told Fox Sports Detroit. "I thought the crowd was great tonight, gave us a lot of energy, and I thought we played a pretty good game overall, so it's nice to get the win for sure."
Nyquist, playing his first NHL game this season, got his first goal off a rebound when he slid a forehand past goalie Justin Peters. Henrik Zetterberg had created a turnover and left a touch pass for a hard shot by Jonathan Ericsson that kicked back to Nyquist at Peters' left.
"[Zetterberg] made a nice play on the wall, on the forecheck and got it to [Ericsson], who shot it," Nyquist said. "Kind of a lucky bounce, just laying around for me, I just banged it, I think five-hole."
In the third period, Nyquist picked up a clear by Zetterberg at the offensive blue line, skated in alone on Peters, deked and slid a backhand into the net with 4:02 to go.
"[It] was a real good head's up play by [Zetterberg], just chipping it out," Nyquist said. "He knew I was coming for a breakaway, so it was nice to get that goal as well."
Nyquist, 24, had been playing for Grand Rapids of the American Hockey League. The 121st pick of the 2008 NHL Draft played 18 games for the Red Wings in 2011-12 and 22 last season.
"He comes in with a lot of energy, a lot of confidence, scores right away on the first shift, scores on his breakaway," said Zetterberg, who assisted on three of the four goals. "It's fun to see. He's probably been waiting for this day ever since training camp. It's nice to see him take advantage right away."
Detroit broke a 2-2 tie in the third period with 15 seconds to go in a two-minute, 5-on-3 power play. A shot by Niklas Kronwall ping-ponged away from the net toward Hurricanes forward Jordan Staal, who kicked it into his own net at 8:29.
Detroit did not allow a goal during a Carolina 5-on-3 of 1:36 near the end of the second period.
"We were right there. ... We had a 5-on-3 that we had the chance to go up," Hurricanes defenseman Andrej Sekera said, according to The Associated Press. "We didn't score and they scored on theirs."
Detroit held a 2-0 lead, but Sekera evened the game at 12:51 of the second period. He picked up a loose puck in the neutral zone, skated around Red Wings forward Todd Bertuzzi and had his shot tip off defenseman Kronwall's stick past Jonas Gustavsson's glove side.
It came 7:02 after defenseman Brett Bellemore got Carolina on the scoreboard with his first NHL goal. Bellemore, a 25-year-old playing his 26th NHL game, finished a give-and-go with Radek Dvorak off the right-wing boards. It was Dvorak's first assist since Feb. 24, 2012, a span of 36 games (17 this season).
Sekera scored a shorthanded goal with 16 seconds to go for the final score.
Detroit took a season-high 47 shots, 18 in the third period. Carolina finished with 27.
"That's what we wanted to do, we wanted to put a lot of pucks on net, and I thought we did that tonight," Nyquist said. "Good things happen when you do that."
The Red Wings took their two-goal lead when Darren Helm scored 2:37 into the second period. Justin Abdelkader was cycling on the right-wing boards, changed direction and headed toward the net. His shot popped back to Helm, who lifted a forehand from the slot over Peters.
Hurricanes forward Jeff Skinner played for the first time since Oct. 24 after he missed 11 games with an upper-body injury. Carolina played its second game without Alexander Semin, who's out with a concussion.
Brendan Shanahan, the NHL senior vice president of player safety and hockey operations, dropped the puck for the ceremonial opening faceoff. Shanahan, who played for the Red Wings from 1996-2006 and was a member of three Stanley Cup-winning teams, was honored before the game for his recent induction into the Hockey Hall of Fame.