“I told everyone I would keep trying to score and I did,” Shesterkin said.
Shesterkin became the 18th goalie to score a goal, and it's the 21st goalie goal in NHL history. He capped the scoring with an unassisted goal into the empty net at 17:26 of the third period for the 5-1 final.
Pavel Dorofeyev scored a goal and picked up his first two points with the Rangers (1-1-0). Mika Zibanejad also had a goal and an assist, Adam Fox had two assists and Eeli Tolvanen scored his first goal with the Rangers.
The Rangers (1-1-0) were 7-for-7 on the penalty kill and 1-for-2 on the power play.
It's a reversal from last season, when the Rangers were shut out in their first three home games and five of their first seven, going 0-6-1.
New York lost 3-0 against the Boston Bruins at TD Garden on Tuesday.
Nikita Kucherov scored and Andrei Vasilevskiy made 20 saves for the Lightning (0-1-0), who lost their season opener.
Zibanejad gave the Rangers a 1-0 lead at 5:34 of the first period on the power play. Dorofeyev's shot from the right face-off circle hit the crossbar and bounced into the left circle to Zibanejad, who shot it in.
New York didn't score at home last season until 57 seconds into its fourth game.
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 on June 26 and signed him to a seven-year, $77 million contract ($11 million average annual value). He led Vegas with 37 goals last season.
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.
