T.J. Oshie had a goal and an assist in the win, which was the second in three games for the Blues and the 17th shutout of Halak's career.
Vladimir Sobotka and Chris Stewart also scored for the Blues, who were outshot by the visiting Blackhawks 29-23. St. Louis scored one goal in each period.
Stewart's second-period score came on a power play, an aspect of the Blues' game Hitchcock had vowed to improve when he was hired Sunday to replace Davis Payne.
Corey Crawford gave up all three goals in the Blackhawks' third consecutive loss.
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