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.
And after witnessing the 21-year-old's strike, Cox tipped Palmer, who scored 25 times in the Conference South last season, for a bright future.
"I don't want to put too much pressure on the kid as he's got to get up to full fitness and he has a lot of learning to do," said Cox.
"But if he carries on the way he has started he has an excellent future ahead of him and not just with Mansfield but a career in football.
"He is an absolute dream to work with, as honest as the day is long and more importantly wants to be a top professional footballer.
"We had this monkey on our back where we hadn't scored a goal, but you saw what it meant to the players and more importantly to the fans and I'm glad that one is out the way.
"I thought the front two before Ollie came on did really well. It's nice to get a monkey off the back and more importantly get the three points and keep a clean sheet."