O'Reilly's shootout goal came in the third round after he had tied the game midway through the third period.
Kevin Shattenkirk gave St. Louis the lead with a power-play goal in the opening period, and Milan Hejduk tied it with 6:05 to go in the stanza.
Patrik Berglund scored for St. Louis in the second.
Varlamov made 35 saves to get the win. He has allowed two goals in each of his last four starts and has gone 2-2 in those games.
Jaroslav Halak took the loss while stopping 33 shots.
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