The win put the defending Stanley Cup champions on the verge of surviving the first round, leading 3-2 heading to Game 6 in Los Angeles on Saturday.
Alex Pietrangelo scored on a wrist shot from the point with 44.1 seconds remaining in regulation and goalie Brian Elliott off for an extra attacker. That forced overtime for the second time in the series.
The Kings' Justin Williams scored in the final minute of Game 1, also in St. Louis.
Jeff Carter scored in the opening minute of the second and third periods for the Kings, who became the first road team to win in the series.
It's hard to walk into that locker room and look those guys in the eye when they've played -- clearly, that was our best game we've played in the series -- and I thought we deserved a better fate tonight.
— Tampa Bay Lightning coach Jon Cooper on his team's 3-2 loss to the Canadiens in Game 3 on Sunday