J.T. Miller, Alexis Lafrenière and Will Borgen also scored for the Rangers, and Artemi Panarin had four assists. Igor Shesterkin finished with 31 saves as New York improved to 8-1-1 on the road.
Scott Sabourin, Zemgus Girgensons and Jake Guentzel scored for the Lightning, who had won six of their previous seven games. Andrei Vasilevskiy gave up five goals on 13 shots through two periods and Jonas Johansson had six saves in the third.
The Rangers, coming off a 6-3 win against Nashville on Monday night for their first home win, finished with their highest scoring total of the season.
Trocheck pushed New York’s lead to 5-3 with 1:41 remaining in the second period as he got a cross-ice pass from Panarin and fired it in from the right circle after Vasilevskiy fell while sliding from his right to his left.
Trocheck scored again at 5:03 of the third from the slot to give the Rangers a three-goal lead, and Cuylle added an empty-netter with 2:11 remaining to cap the scoring.
Cuylle got the Rangers on the scoreboard with a power-play goal at 1:08. Adam Fox’s shot from above the inside edge of the left circle deflected off Lightning defenceman Erik Cernak and then the end boards right to Cuylle on the left doorstep for a quick goal.
The Rangers doubled the Lead at 3:31 when Mika Zibanejad’s pass went off Miller’s skate and in.
Sabourin pulled the Lightning to 2-1 as he beat Shesterkin from the left circle at 6:04 for his first, and Lafrenière scored from the right circle 1:04 later to restore New York’s two-goal lead.
Girgensons pulled Tampa Bay back within one with 6:01 left in the first. Oliver Bjoristrand’s shot from the slot hit Yanni Gourde in front and Girgensons put the loose puck in.
Borgen capped the Rangers’ four-goal opening period with 3:40 remaining and Guentzel scored a short-handed goal 1:12 later to pull the Lightning to 4-3.
Rangers: Play at Columbus on Saturday.
Lightning: Visit Florida on Saturday.
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