Not only did the Predators defeat the defending Stanley Cup champs, who are also second in the Western Conference, they also beat the conference's top team, the San Jose Sharks, on Thursday night, and Los Angeles on Saturday night in overtime.
"This crew deserves a lot of credit for this week," coach Barry Trotz said.
It also was Nashville's fourth game in six days.
"We're in the fight of our lives for the playoffs," Sullivan said. "We've still got a lot of work to do."
Henrik Zetterberg, Mikael Samuelsson and Johan Franzen scored for Detroit, which lost is second straight and third in four games. Pavel Datsyuk and Nicklas Lidstrom each had two assists, and Chris Osgood stopped 22 shots.
"The bottom line is we got to find ways to win games and we're in it together," Osgood said.
Suter beat him with a slap shot from the top of the left circle for his seventh goal, just 29 seconds after Chris Chelios took a cross-checking penalty.
Sullivan made the play as he took the puck off the left boards inside the blue line and headed into the middle of the ice, but he then passed it back to Suter at the left point, who moved in and blasted a shot high on the far side.
"Sully made a great play coming off the wall," Suter said. "I thought he was going to go to the other point. But he passed it back to me. I moved in and was looking for someone to pass to. I saw Bonkie [Radek Bonk] in front of the net and then I just tried to go far side."
Franzen tied it at 3 with 5:25 left with a wrist shot from the high slot for his 32nd goal.
Goals by Ward and Smithson 20 seconds apart gave Nashville a 3-2 lead.
Ward tied it at 2, putting in a rebound for his 17th goal with 8:25 left in the third period. Smithson took a pass from Cal O'Reilly and beat Osgood from the bottom of the left circle. It was Smithson's fourth goal.
"A point shot. I went to the net to try and create a traffic," Ward said. " And I was able to chip the rebound in."
"We actually checked our own guy in the neutral zone on the one," Red Wings coach Mike Babcock said. "We found a way to lose the game. That drives you crazy, especially this time of year. It's the time of year you want to be ramping it up."
Samuelsson's power-play goal gave Detroit a 2-1 lead with 4:47 left in the second period. It came with the Red Wings holding a two-man advantage. Samuelsson put in a one-time shot from the left circle for his 19th goal.
Sullivan opened the scoring with a power-play goal 2:25 into the game for his 10th goal. Zetterberg tied it with his 30th goal, a short-handed effort at 4:29.
"I thought we were gassed after the first period," Trotz said. "But the guys just sucked it up."
- Detroit D Brian Rafalski missed the game because of a sore groin.
- Nashville was without captain and C Jason Arnott (concussion), C David Legwand (cheekbone), D Wade Belak (lower body) and LW Ryan Jones (upper body).
- An apparent goal Franzen with 1:15 left in the first was reversed because it was ruled the he kicked it in.
- Predators' RW Martin Erat was helped off the ice after getting hit in the left leg by teammate Shea Weber's slap shot a little less than 1:30 into the second period and didn't return. Trotz said Erat likely had a fractured leg and would be out for at least four weeks.
- Zetterberg's goal was his 400th career point.