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.
Jamie McAllister went straight back into the City side after a six-week injury absence as one of two changes made by Robins boss Keith Millen.
Sidelined by a broken rib, left-back McAllister replaced injured on-loan defender Steven Caulker after proving his fitness in a reserve-team game this week.
Lee Johnson took over from thigh strain victim Kalifa Cisse in the midfield holding role, while Cole Skuse and Jon Stead were named on the bench after 10 weeks out injured.
Mustapha Dumbuya replaced Sam Hird in defence in the only change made by Rovers.
The game took a long time to get going. David James made a routine save from Simon Gillett in the opening minute and Gary Woods did likewise at the opposite end from a Lewin Nyatanga header after 13 minutes.
James was called into more athletic action just before the half-hour mark when he came out to block Franck Moussa 's shot.
The former England keeper was then properly tested in the 35th minute when he had to tip Gillett's 12-yard shot round a post.
At the other end, Woods produced a similar save to keep out Albert Adomah's effort just before the break.
Maynard threatened just before the hour mark and on-loan defender Jordan Spence sent an effort narrowly wide soon afterwards.
City continued to apply pressure and forced Woods into a couple of saves in quick succession midway through the second half. First Woods parried a shot from Spence and then he blocked a volley from Adomah.
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.
Millen's men shut-up shop after that and saw out the remaining quarter of an hour to secure a fourth straight home win.
Meanwhile, Doncaster have won only one of their last 11 matches and could yet be dragged into the relegation mire.