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.
Hopefully [Jonathan] has a year that pushes us, that is the organization, to speak to him next summer and looking for [a long-term contract]. Because the thought process is trying to get a goaltender, and hopefully it's Jonathan, to be the stabilizer for this franchise.
— Maple Leafs GM Lou Lamoriello on goaltender Jonathan Bernier