Kidsgrove made most of the running in the opening 25 minutes and Matt Haddrell wasted a good chance when sending a header from point-blank range to the keeper.
At the other end, Steve Intihar made a one-handed save from Bay's Michael McGrath but it was against the run of play when the visitors from the Premier Division took the lead on 37 minutes.
Craig Dove and Nick Ward collided in defending a long ball by Anthony Sheehan and Alex Titchener netted with a half-volley into the far corner.
Dave Walker hit the foot of a post on the restart but Kidsgrove went two down on the hour when Joseph O'Sullivan's shot went through the legs of Intihar.
Shotton and Walker wasted chances to put Kidsgrove back in the game while Intihar saved well from Ian Sheridan.