Brian Stock was the League One side's hero with a 20-yard drive which took a cruel deflection to leave Carl Muggleton stranded.
The goal cancelled out talisman striker Richie Barker's 23rd minute penalty, just when the Stags thought they had survived late pressure.
Doncaster, who dominated the latter stages of both halves, looked to have squandered their best chance of forcing a replay six minutes from time when James Coppinger's cross was headed inches wide from point-blank range by Jason Price.
In-form Rovers forced the first chance, Lewis Guy firing straight at the keeper after a forceful run to the edge of the area.
But the Stags almost found an early breakthrough too. Jake Buxton threaded a pass to Michael Boulding and he was only stopped by keeper Ben Smith's punch.
Smith, replacing on-loan veteran Neil Sullivan, then watched as Giles Coke blasted a 30-yard drive narrowly over and Matt Hamshaw curled inches wide from inside the area.
But the young keeper had no chance when the home side, looking more like the team from the higher division, snatched a deserved 23rd minute lead.
Gareth Roberts inexplicably handled as he tried to control a bouncing ball and Barker calmly slotted the penalty into the bottom corner.
However, the strike galvanised Doncaster who came storming back in search of a quick equaliser.
Price and Sean Thornton both had drives blocked by desperate defending and then the visitors had two loud appeals for a penalty of their own turned down.
Guy twice burst clear of the last defender as he raced into the area, but even though he appeared to be tripped by first Alex John-Baptiste and then Jon Hjelde the referee waved away Rovers' spot-kick claims.
In first-half injury-time Rovers went even closer, Mark McCammon's flick from a Thornton corner just evading Jonathan Forte as he tried to touch in from close range.
Rovers, now dominating through their slick passing, almost fell further behind, though, a minute into the second half when Simon Brown's cross-shot was beaten at the near post by the alert Smith.
Mansfield were almost made to pay for missed chances when the unmarked Price flashed an 18-yard shot over and referee Mick Russell waved away three more penalty appeals.
Then Barker came within an inch of doubling the lead in the 68th minute. The striker rounded Smith, but rolled his finish from the tightest of angles across the face of the empty net.
And Mansfield were made to suffer when Rovers struck in injury-time.