Lafreniere has seven goals during the streak that started with a hat trick against the New York Islanders on Oct. 6.
Will Cuylle and Mika Zibanejad scored, Tye Kartye assisted on both of Lafreniere's goals, and Dylan Garand made 25 saves.
The Rangers (6-2-0) improved to 5-0-0 at home, all regulation wins. They had nine regulation wins at home all of last season and didn't get their fifth at home last season until Dec. 20 -- in their 18th home game of the season.
The last time New York won at least five in a row at home to start a season was 1990-91 (8-0-0).
Zemgus Girgensons scored, and Andrei Vasilevskiy made 19 saves for the Lightning (4-2-0), who had their four-game winning streak end. Tampa Bay's two losses this season have both come against the Rangers at the Garden.
Lafreniere scored 62 seconds into the game to give the Rangers a 1-0 lead.
Kartye had a strong forecheck to win the puck from Victor Hedman. He tried to get it to the front of the net, but the puck got shoved back behind the goal. Kartye chased it down and sent another pass to the front, this one getting to Lafreniere's stick in the slot.
Kartye and Lafreniere connected again at 12:47 to extend New York's lead to 2-0.
Kartye flipped a stretch pass from the Rangers' defensive zone up to Lafreniere, who got ahead of Hedman and Anthony Cirelli, caught the puck in stride, protected it on his forehand and scored.
Girgensons cut it to 2-1 at 17:26, keeping it on a 2-on-1 with Ilya Mikheyev to his left and scoring over Garand's left shoulder.
Cuylle scored on a deflection from between the hash marks at 18:26 of the second to give the Rangers a 3-1 lead.
Zibanejad stole the puck from Brayden Point and scored a short-handed empty-net goal at 17:16 for the 4-1 final.
