The Lightning were on a power play 1:10 into the middle period when Hedman scored to forge a 1-1 tie.
On the ensuing face-off, Radko Gudas found a streaking Kucherov alone at center ice and the Tampa Bay winger broke in on Henrik Lundqvist to score the decisive goal.
Ben Bishop saved 33-of-34 shots as Tampa Bay finished a three-game road trip with a 2-1-0 mark.
Brad Richards scored and Lundqvist finished with 23 saves for the Rangers, whose three-game winning streak was ended.
Boston schools pull out free condoms over wrapping complaints
NBC reportedly holds celebs hostage to Jimmy Fallon's show