Calgary trailed 3-0 when Mason Raymond broke up the shutout early in the third period. Josh Jooris narrowed the lead with a goal in his NHL debut, scoring off a pass from Mark Giordano with over eight minutes left.
The Flames, five games into a six-game road trip after opening at home, threatened to the end.
Calvert had an assist to go with his goal. Mark Letestu had two assists as the Blue Jackets wrapped up a three-game homestand.
The Blue Jackets were 4 for 4 on the penalty kill to stay perfect this season in nine attempts.
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