Detroit, which has missed the playoffs in each of the last six seasons, moved into a third-place tie in the Atlantic Division with Tampa Bay.
After Stamkos extended his point streak to 11 game with a goal at 14:52 of the third with Vasilevskiy on the bench, Erne put the Red Wings up 3-1 with 3:38 remaining, scoring into an empty net.
Point got another goal with an extra skater at 17:34, before Perron picked up another empty-netter in the final minute.
Detroit coach Derek Lalonde faced the Lightning for the first time after being an assistant coach with Tampa Bay from 2018-22, which included Stanley Cup championships in 2020 and 2021
"Really cool being back," Lalonde said.
Lalonde replaced Jeff Blashill, who is now a Tampa Bay assistant coach.
Rasmussen got past the Lightning defense to open the scoring 1:54 into the second from the slot off a nifty pass from Dominik Kubalik.
Filip Hronek got the second assist on Rasmussen's goal to set a franchise record with a season-opening 11-game road point streak. He is the second NHL defenseman to have an 11-game road point streak to start a season, joining Edmonton's Paul Coffey (12 games in 1986-87).
Berggren made it 2-0 at 5:59 of the third with a slot redirection of Olli Maatta's shot. The rookie has three goals and eight points in 12 games.
Kubalik had his breakaway shot stopped by Vasilevskiy and Jordan Oesterle hit the post as Detroit outshot the Lightning 12-11 during a scoreless first. Husso made strong saves on Nikita Kucherov and Vladislav Namestnikov.
Tampa Bay committed 11 giveaways, while the Red Wings had none in the opening 20 minutes. The total was 14-0 after two periods.
Tampa Bay's Nick Paul had his short-handed shot hit the post with 45 seconds left in the second.
Tampa Bay had its 10-game power-play goal streak end after going 0 for 2. The Red Wings failed to score on its lone chance.
Red Wings: C Robby Fabbri (knee) and LW Elmer Söderblom (lower body) took part in an extensive on-ice workout following the morning skate.
Red Wings: Play Thursday night at Florida.
Lightning: Host Nashville on Thursday night.
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- Kyle Rehman
- Pierre Lambert
- Jonny Murray
- Michel Cormier