In the latest installment of a riveting Western Conference quarterfinal series, Los Angeles gave up the lead late in regulation and then came through with the clutch goal that put the Kings in front three games to two.
Los Angeles will have a chance to close out the series when it hosts Game 6 on Friday.
Each of the first five games in the series has been decided by one goal with two of them going to overtime.
Alexander Steen lifted St. Louis into a 1-1 tie 6:46 into the second period and Alex Pietrangelo kept the Blues' hopes alive by knotting things up with 15 seconds left in the third.
The win went to Jonathan Quick, who made 34 saves. Elliott stopped 22-of-25 shots.