In a match that followed a similar pattern to the one at Liverpool, the visitors led at half time but conceded three times in the second period.
Mitchell drew first blood in the dying embers of the first half when pouncing on the rebound after his shot struck the woodwork, following good work by Ryan McConnell and Devonte Redmond.
Mitchell, playing in attack due to the absence of James Wilson and Ashley Fletcher, had earlier gone close to scoring as Callum Williams denied the England Schoolboys star and Jordan Thompson. Dean Henderson also pulled off one fabulous save to thwart Brice Wassi.
Rovers came out strongly after the break and responded well to the setback of conceding with Dean Rittenberg swiftly equalising by latching onto a Jordan Preston flick-on within two minutes. Just past the hour mark, Preston netted himself with a well-struck effort from distance.
United poured forward in search of a leveller but Thompson's shot inches wide proved to be the best attempt as, with 10 minutes left, substitute Modou Cham drilled home a loose ball from 18 yards out.
United: Henderson; Evans, McConnell, Borthwick-Jackson, Barber (Ioannou 80); Harrop, Goss, Willock, Redmond, Thompson (Riley 75); Mitchell (Rashford 80). Sub not used: O.Byrne.