Exeter led 26-0 at halftime, but two yellow cards either side of the interval aided Saints' comeback to become the first winners at Sandy Park since October 2022.
The visitors' Ollie Sleightholme scored a hat-trick, but his efforts were undone when Rus Tuima scored with eight minutes remaining.
However, the Saints gained a late line-out, and Hutchinson went over a few phases later.
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