After the Rangers' Brad Richards beat Tampa Bay goaltender Mathieu Garon to the left side but hit the post with his shot, Moore swooped down the right side and ripped a wrister under the glove of Henrik Lundqvist for the game- winning goal.
Marian Gaborik was then denied by Garon's sliding pad save to snap the Lightning's five-game losing streak.
Moore also netted the game-tying goal late in the third period.
The Rangers tallied the lone goal of the first period when Callahan took advantage of a power play with a shot from the left circle that sailed over Garon's right shoulder and under the crossbar on the short side at the 7:49 mark.
Ryan Malone tied the game up 11:23 into the second period by intercepting a Rangers pass at the New York blueline, racing into the slot and sending a wrister past the glove of Lundqvist.
But two minutes later, the Rangers streaked down the ice on a shorthanded rush and Anisimov stuffed home a shot from in front off a Michael Del Zotto pass to put New York back on top.
After the goal, Anisimov made a motion with his stick like he was shooting a rifle towards Tampa Bay's Vincent Lecavalier, causing a huge scrum that resulted in multiple penalties.
Tampa Bay came out of the melee with a two-man advantage but could not convert on the opportunity and New York took a 2-1 lead into the second intermission.
It looked as if the game would end with that score or New York would extend its lead as the Lightning took a late boarding penalty in the third period.
But Moore raced into the offensive zone and sent a backhander from the left side that deflected off the stick of a Rangers player and fluttered over Lundqvist's shoulder on the short side for a short-handed goal with 1:57 left in regulation to force overtime.
The Rangers dominated the extra period, outshooting the Lightning 6-1 and even hitting the post, but New York could not solve Garon to make the shootout necessary.
Tampa Bay's Martin St. Louis was hit in the face with a puck during Thursday's morning practice and will be out indefinitely with facial and nasal fractures...Garon stopped 28-of-30 shots, while Lundqvist made 29 saves...Richards had an assist to extend his point streak to seven games.