The game looked to be heading for an uninspiring goalless draw until Maynard intervened in the 76th minute.
It was the antithesis of last season`s corresponding fixture which saw Rovers triumph 5-2.
The south Yorkshire club edged the first half but City were the more threatening after the break and Maynard broke the deadlock with a 12-yard finish.
Albert Adomah was the architect when City made the breakthrough. He beat his man down the right-hand side and crossed for Maynard to convert from 12 yards out.
Source: PA
Source: PA