After scoring the only goal of the game in their Easter Monday triumph at Sheffield Wednesday, Maynard tapped home from close range in the 84th minute to extend the Robins’ unbeaten run under Millen to six matches and keep their faint play-off hopes alive.
By contrast, sixth-placed Swansea are now looking anxiously over their shoulders towards Blackpool, who are just two points adrift with three games remaining.
After drawing 17 games apiece this season another stalemate appeared to be on the cards for the Robins and the Swans.
Maynard had his side’s best chance of the opening period, while Swansea striker Gorka Pintado fluffed a one-on-one opportunity.
The sides seemed to be heading for their fifth goalless draw in six meetings, but Maynard had other ideas.
Swansea made three changes to the side that brushed aside Scunthorpe 3-0 on Easter Monday. Two of those were in midfield, with Darren Pratley and Andrea Orlandi consigning Mark Gower and Guillem Bauza to the bench, while Pintado replaced Shefki Kuqi in attack.
Bristol City were unchanged, with young Irish goalkeeper Stephen Henderson making his home debut.
The visitors made much of the early running, and Orlandi and Pratley combined on the edge of the Bristol box before the latter’s effort was deflected wide by Paul Hartley in the third minute.
The Swans’ early momentum was checked when Angel Rangel was forced off with a thigh problem, allowing Albert Serran to make his return from a three-month injury lay-off.
In the 12th minute, Robins full-back Bradley Orr burst into the penalty area to collect Jamal Campbell-Ryce’s crossfield ball, only to fire high and wide.
Bristol started to play more like a side flushed with the confidence of an extended unbeaten run, and Maynard pulled narrowly wide following a 23rd-minute melee in the Swansea box.
Paulo Sousa’s side responded by fashioning two inviting openings for themselves. First Orlandi’s 30-yard free-kick dropped tantalisingly wide, before Pintado shot straight at Henderson when clean through on goal.
Campbell-Ryce was denied the opener shortly before the interval when Swansea keeper Dorus de Vries grasped his 20-yard drive at the second attempt.
After half-time, the game progressed serenely without incident and it was left to Robins substitute Ivan Sproule to inject some excitement when he fizzed a 67th-minute shot across the face of De Vries’ goal and narrowly wide shortly after coming on.
Bristol gradually increased the pressure and broke through six minutes from time when Sproule got the better of David Edgar on the right wing before finding the unmarked Maynard, who applied the finish.
Sproule almost fashioned a second in the final minute when fellow substitute David Clarkson headed his cross wide.