Eller made a couple moves before beating Ben Bishop on the Canadiens' first shootout attempt.
Tomas Plekanec scored in regulation for Montreal. Price stopped 23 shots through overtime.
Martin St. Louis scored for the Lightning, who also got 20 saves in regulation and overtime from Bishop. Tampa Bay had its five-game winning streak end.
Plekanec opened the scoring, beating Bishop with a backhander 5:15 into the second period. It was his 169th goal, which moved him past Shayne Corson and Gilles Tremblay into sole possession of 31st place on Montreal's career list.
St. Louis tied it from the top of the right circle on Tampa Bay's sixth shot of the game at 11:43 of the second. The Lightning captain has a goal in four straight games and eight points during a six-game points streak.
Montreal had the first six shots, including an in-close backhander by Daniel Briere and a shot from the left circle from Plekanec. Nate Thompson got Tampa Bay's first shot 11 minutes into to the game, a backhander stopped by Price.
The Canadiens outshot Tampa Bay 15-9 through two periods. The Lightning had nine of the 11 shots during the first 9 minutes of the third, and 14-6 overall edge in the period.
Price made a save midway through the third on Radko Gudas' backhander.
NOTES: St. Louis has 950 points, which is two behind Rick Tocchet, who is 89th on the NHL list. ... Montreal D Josh Gorges returned after missing a game because of a lower-body injury. ... Tampa Bay D Eric Brewer and D Sami Salo both sat out with upper-body injuries. ... Canadiens rookie LW Michael Bournival, pointless over his previous 13 games, didn't play.