The Stags led when Chris Clements curled in a free-kick after Stevenage's Michael Tonge had earlier hit the post.
James Baxendale doubled the lead on his debut, after signing on loan from Walsall, by firing in from 20 yards.
Armand Gnanduillet netted a consolation for Stevenage with the last kick of the game, tapping in from eight yards for his sixth goal of the season.
Stevenage manager Teddy Sheringham told BBC Three Counties Radio:
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"It was disappointing to go in a goal down when you are on the front foot and feeling good about the way things are going."
On Mansfield's penalty: "It was seen by everyone in the ground that it's hit his hand. From my view if you're giving it last week then that's a stone-wall penalty, without a doubt. We're not getting that rub of the green at the moment.
"Was it a free kick for their goal? I don't know. The man has skipped over a challenging leg, I don't know whether it's much of a free kick myself. It's given, and he puts it in the top corner."
Mansfield achieved their first league win since November as they saw off struggling Stevenage.