Dorofeyev extended it to 2-0 with his first goal with the Rangers at 10:51, a one-timer from the inside edge of the left circle off a pass from Oliver Bjorkstrand.
The Rangers acquired Dorofeyev from the Vegas Golden Knights for three draft picks on June 26 before signing him to a seven-year, $77 million contract ($11 million average annual value) on June 30. He led Vegas with 37 goals last season.
"Everyone wants to score; that's what we're getting paid for, just part of our job, you know," Dorofeyev said. "It obviously feels great when you get it sooner than later."
Tolvanen made it 3-0 at 1:40 of the second period. He took the puck off the right-wing wall and sent it into the slot for Will Cuylle. It never touched Cuylle and instead went in off Tampa Bay defenseman Charle-Edouard D'Astous' right leg.
Kucherov scored with a backhanded pass that went in off Noah Laba at 7:59 to it to 3-1, but Gabe Perreault got it back for the Rangers with a redirection goal, the puck going off his stick and skate at 16:01, to make it 4-1.
To Cooper's point on lack of pushback, the Lightning managed just six shots on goal in the third period despite being on the power play for nearly seven minutes and pulling Vasilevskiy twice to create a 6-on-4 power-play advantage midway through the period.
"We did a great job," Shesterkin said. "PK was awesome. We did an unbelievable job, especially against Tampa. It's one of the best power-play teams, against 'Kuch.' We played the right way, everyone did what they need to do, and we just worked for the team."
NOTES: The win was Shesterkin's 100th at Madison Square Garden in 177 games. He's the fourth-fastest goalie in Rangers history to record 100 home wins after Mike Richter (172 games), Eddie Giacomin (173) and Henrik Lundqvist (176). … Zibanejad recorded the first goal of the game for the 60th time with the Rangers and surpassed Chris Kreider (59) for the second-most instances in franchise history, behind only Rod Gilbert (72).