The summer signing from Havant & Waterlooville struck nine minutes after coming off the bench when he tapped in after Chris Clements' free-kick was deflected against the post.
It was Palmer's first goal for the club and came 12 minutes from time after Wycombe goalkeeper Matt Ingram had earlier denied James Jennings.
Jennings came close early on for the Stags with a beautifully struck volley, forcing Ingram into a fine save.
Paris Cowan-Hall finally produced some class for Wycombe by first setting up midfielder Josh Scowen and then blasting a shot from long range but both were superbly saved by Mansfield keeper Alan Marriott.
Cowan-Hall was stretchered off in the 65th minute after a miss-timed tackle from midfielder Jamie McGuire who was booked for the challenge.
However, Mansfield took the lead their play deserved when substitute Palmer poked in from close range after Martin Riley's header rebounded off the post with Jo Kuffour then heading wide at the death for Wycombe.