The Cubs threatened again in the seventh when Michael Busch singled and Michael Conforto doubled, putting runners on second and third.
Dobbins escaped the jam by striking out Dansby Swanson and forcing Nico Hoerner into a groundout.
St. Louis attempted to respond in the bottom of the seventh when Wetherholt walked, stole second, and reached third on a wild pitch, followed by an Herrera walk.
However, Walker hit into a double play turned by Bregman.
Dobbins successfully managed the final two innings, concluding his appearance with four strikeouts and no walks to secure his first career save.
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The St. Louis Cardinals remain at Busch Stadium on Monday evening to host the Texas Rangers, with Michael McGreevy scheduled to start against Texas pitcher Jacob deGrom.
