Stags' early dominance paid off in the 26th minute as Lucas Akins came in from the right and played a perfect ball into the path of Rhys Oates to sweep home first time inside the left-hand post.
Liam Roberts kept out Robbie Gott's 25-yard drive in the 35th minute while Louis Reed whistled a rising shot over the Doncaster crossbar in first-half stoppage time.
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An energised Rovers levelled four minutes after the break when Gotts' through ball cut Stags open down the centre and Jack Senior got there first to drill the ball low past Roberts.
And Rovers should have gone ahead in the 59th minute as Jamie Sterry's perfect low cross to the far post saw substitute Brandon Hanlan fire wide from eight yards.
Stags rallied and Michael Smith shot inches past the upright after 64 minutes and Oates went even closer with a superb overhead-kick shortly afterwards.
Mansfield deservedly won it in the 74th minute with a training-ground free-kick.
Regan Hendry, wide on the right, rolled the ball into the path of Reed, whose low, first-time finish found the bottom-left corner from the edge of the box.
