With the Kings on a power play 13:26 into the extra frame, the Los Angeles goalie skated behind the net to get control of a loose puck on a seemingly routine play.
But Steen quickly skated in from his right and Quick tried keep the disc away from him by shooting it off the end boards. Steen, however, intercepted it, then stuffed the puck into an empty net for a stunning game-winner.
The bizarre tally was Steen's second goal of the game and gave the fourth-seeded Blues a 1-0 lead to begin their Western Conference quarterfinal series against the fifth-seeded defending Stanley Cup champions.
Brian Elliott made 28 saves to earn the victory.
Justin Williams forced overtime when he scored with 31.6 seconds left for Los Angeles.
Quick finished with 40 saves in the loss.