After Nikita Kucherov opened the scoring with a power-play marker 2:41 into the first period, the Rangers forged ahead on goals from K’Andre Miller and Kaapo Kakko.
Mika Zibanejad beat Andrei Vasilevskiy blocker-side early in the second to put New York up by a pair.
Nick Paul stuffed a shot home with 2:02 left in regulation to draw the Lightning within one, but the defending champions could not come up with the equalizer and now head back to Tampa down two games.
The Lightning had won 17 straight games in the postseason following a loss since the beginning of the 2020 Stanley Cup Playoffs. That streak is no more.
Former Florida Panthers forward Frank Vatrano had the primary assist on Miller’s goal.