Price sprawled to the ice to poke-check the puck away from NHL goal leader Steven Stamkos and then turned aside Martin St. Louis' 360-degree turn attempt.
After Simon Gagne scored on Tampa Bay's third attempt to extend the shootout, Michael Cammalleri snapped a shot past Dwayne Roloson. Price then stopped Teddy Purcell's deke with a glove save to secure the win.
Price stopped 35 shots, and Brian Gionta and Andrei Kostitsyn had goals in regulation for sixth-place Montreal, which has 85 points — three fewer than fifth-place Tampa Bay in the Eastern Conference playoff race.
In Nashville, Shea Weber scored a power-play goal with 1:23 left in overtime to lift Nashville past Boston.
Steven Kampfer was penalized for holding to set up the power play. Weber scored off a pass from Ryan Suter and Nashville won for the first time in overtime this season.
Nashville's David Legwand tied it at 10:56 of the third period. Colin Wilson and Sergei Kostitsyn each had a goal and an assist for the Predators.
Patrice Bergeron, Tyler Seguin and David Krejci also scored for Boston.
In Dallas, Jamie Benn extended his point streak to nine games with a goal and an assist that helped Dallas rout Chicago.
Kari Lehtonen made 23 saves for his 16th NHL shutout, while Steve Ott, Brenden Morrow, Jason Williams and Brandon Segal added goals for the Stars.
In Columbus, recently recalled goalie Joey MacDonald made 37 saves for only his second NHL shutout as Detroit downed Columbus.
Drew Miller and Valtteri Filppula scored first-period goals for the Red Wings, who won their fourth in a row to give Mike Babcock his 300th win with Detroit. He is the third NHL coach to win 300 games with a team over a six-season span, joining Montreal's Scotty Bowman (321) and Edmonton's Glen Sather (307).
MacDonald joined the team on Wednesday while Chris Osgood continues to rehab from hernia surgery. He preserved the shutout with a late goal-line glove save that withstood a video review.
The Blue Jackets have lost nine of 10.
In San Jose, Patrick Marleau got a fortunate bounce on his tiebreaking, power-play goal early in the third period and San Jose went on to beat Minnesota.
Logan Couture and Jamal Mayers also scored for the Sharks, who stayed atop the tight Pacific Division standings.
John Madden and Kyle Brodziak scored for Minnesota, which finished an 0-4 road trip that put a severe dent into their playoff hopes.
In Atlanta, Rob Schremp and Blake Wheeler scored in the shootout to give Atlanta victory over Philadelphia after squandering three leads.
The Thrashers closed within four points of eighth-placed Buffalo in the playoff race with their second win in six days over the Eastern Conference-leading Flyers. Atlanta will face Buffalo on Saturday.
Danny Briere scored two goals, including a tying shot with 1:17 left in regulation for the Flyers.
In Los Angeles, Jaroslav Halak stopped 17 shots for his 14th NHL shutout as St. Louis had a surprisingly easy win over Los Angeles.
Matt D'Agostini had his third career two-goal game for St. Louis, which swept Los Angeles in four games this season.
The loss, which snapped a four-game winning streak by the Kings, dropped them from fourth to fifth in the tight Western Conference.
The Blues, 10 points behind eighth-place Calgary, are likely to miss the playoffs for the fifth time in six years.
Alex Pietrangelo and T.J. Oshie also scored for St. Louis.
In Calgary, Jarome Iginla scored two goals as Calgary sent Colorado to its 10th straight loss.
Rene Bourque and Alex Kotalik had a goal and an assist each for the Flames, who snapped a three-game losing streak and moved back into eighth place in the Western Conference.
Mark Giordano also scored for Calgary.
Mark Olver and Cody McLeod had goals for the Avalanche.
In Edmonton, Ray Whitney and Radim Vrbata each had a goal and an assist as Phoenix beat Edmonton for its fourth straight win.
Lee Stempniak also scored for the Coyotes.
Magnus Paajarvi scored for the Oilers who have lost five straight and are last in the NHL.
In Sunrise, Florida, Sergei Samsonov and Stephen Weiss each had a goal and an assist in Florida's win over Toronto.
Scott Clemmensen stopped 24 shots for his first shutout of the season while Mike Santorelli also scored for the Panthers, whose last 11 victories had been by one goal.
Keaton Ellerby added an empty-netter.
In Ottawa, Chris Neil scored two third-period goals to help Ottawa beat New Jersey for the second time in nine days.
The Devils are six points behind eighth-placed Buffalo for the final Eastern Conference playoff spot.
Ryan Shannon also scored for the Senators.
Brian Rolston scored for the Devils.
