The Senators were down in the tiebreaker in the third round, but Stephane Da Costa kept the game alive with delayed backhander for a score.
In the fifth round, the Blues' Maxim Lapierre missed the net high, and Turris followed by unleashing a long wrist shot that Halak couldn't catch up with, giving the Sens a victory after two straight road losses.
Ottawa rallied from a 3-1 deficit in the third period with goals from Milan Michalek, Turris and Jason Spezza in a 2 1/2-minute span, but then surrendered a goal to Jordan Leopold to knot the score 4-4.
Erik Karlsson scored and added an assist and Spezza provided a pair of assists.
Robin Lehner got the hard-earned the win with 46 saves.
T.J. Oshie logged a goal and an assist for St. Louis, which was coming off a shootout win over Nashville Saturday.