Tampa took the lead early via Pat Maroon in the first period, and extended to 2-0 in the second via Nikita Kutjerov.
In the third, the home team pulled away to 3-0 via Steven Stamkos.
That proved to be enough to secure the second home win in a row, before the match series moves to Manhattan again, where the Rangers took the two initial victories.
Rangers shot the most in the match 34 against 31, but Andrei Vasilevsky had a good day between the posts and held tight until the final sprint.
In the end, the onslaught became too much for the Russian star goalkeeper, when Rangers pressed on with six outfielders against four, after Alex Killorn went on an expulsion in the 16th minute.
But the Rangers did not come any closer than that, and the final result was finally written 4-1 after Ondrej Palat hit one last puck in the open goal with nine seconds left of the match.