The hosts' top scorer pounced with six minutes left to convert substitute Ivan Sproule's superb run and cross.
The late winner lit up an otherwise dour encounter played out in the searing South West heat.
Bristol have now gone six games unbeaten - increasing caretaker manager Keith Millen's chances of succeeding Gary Johnson's in the Ashton Gate hot-seat.
Danny Haynes and Maynard both fired early chances wide for the home side, while Darren Pratley was denied a chance inside the box by Paul Hartley's last-ditch tackle at the other end.
Andrea Orlandi curled a free kick wide of the upright on the half-hour, before an even better chance came the Swans' way.
David Cotterill, a former Bristol City favourite, skipped by Bradley Orr to set the attack on its way. Pratley received his pass on the edge of the box and then helped the ball into the path of Gorka Pintado inside the penalty area.
Pintado advanced on Stephen Henderson's goal, but couldn't manage to guide the ball in, the young City keeper saving well with his body.
After half-time chances dried up with Hartley and Cotterill both wasting free-kick opportunities from dangerous positions.
Millen threw on Sproule and David Clarkson to liven up Bristol's attack and the ploy worked as Swansea tired.
In the end Sproule used his pace to beat marker David Edgar down the right flank. The Northern Irish winger reached the by-line and cut the ball back brilliantly to present Maynard with a simple tap-in.
Clarkson had a chance to seal the win when Sproule's dinked cross landed perfectly for his header, only for the ball to drop wide of the mark.