Jagr, Martin Straka and Michael Nylander each had a goal and two assists for New York, which snapped a two-game losing streak. Brendan Shanahan added a goal and an assist to extend his point streak to seven games.
A night after a lacklustre performance in a 3-1 loss at Pittsburgh, the Rangers applied constant pressure on Tampa Bay goalie Johan Holmqvist, who had won his previous five starts.
The Lightning had won four of five but fell behind 2-0 after two periods and allowed a power-play goal to Shanahan and an even-strength tally by Straka in the third.
Eric Perrin scored with 2:40 left in the game to end Lundqvist's shutout bid.