Roared on by over 3,000 away fans, Nigel Clough's men failed to fire in the first half and fell behind to Branden Horton's seventh minute tap-in.
Mansfield enjoyed the early territory but it was Rovers who took the lead with their first foray forward.
The hosts got in behind and Aidan Barlow unselfishly crossed for left wing back Horton to slot home.
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Stags struggled to muster a response during a largely uneventful first half.
Lapslie did have the ball in the back of the net only to see his smart finish ruled out for offside, while at the other end John-Joe O'Toole made a vital block to keep out Tommy Rowe's goalbound header.
Jordan Bowery replaced Stephen McLaughlin at the break and Mansfield were immediately on the front foot.
George Maris surged into the box and was only denied by an excellent save from Jones.
From the resultant corner Town were level as a near post flick on reached Forrester who gratefully headed home on 48 minutes.
The away end was now in full voice and Doncaster were doing much more defending.
The game was turned on its head on the hour mark when Bowery's shot was palmed away by Jones but Lapslie was on hand to ram home the rebound from a narrow angle.
Town then sliced open a fragile Doncaster team with an incisive move on 71 minutes and Lapslie finished with aplomb.
Rowe fired home from a well-worked free kick for Rovers to set up a nervous finale, but Mansfield held firm to see the game out.
DONCASTER: Jones, Knoyle, Anderson, Rowe, Horton, Hasani, Ravenhill (Blythe 57), Smith, Barlow (Cukur 64), Hiwula, Dodoo. SUBS NOT USED: Dahlberg, Faulkner, Cole, Wilds, Henson, Kuleya.
STAGS: Bishop, Hewitt, O'Toole, Forrester, McLaughlin (Bowery 46), Stirk, Clarke (Charsley 90), Lapslie, Quinn, Maris, Oates (Sinclair). SUBS NOT USED: Shelvey, Burke, Ward, Law, Mason.
REFEREE: Sebastian Stockbridge.