Nesbitt scored twice in regulation for Peoria (12-11-4), while Scott Ford added a goal and an assist.
Robert Mayer made 32 saves in taking the loss.
Peoria led 3-1 after two quick goals early in the third, but Gallagher scored for Hamilton at 4:53 before Chaput tied it at 10:52.
The Rivermen finished the night 1-for-3 on the power play, while the Bulldogs were scoreless in five chances.
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