In the opening chance of the game, Marvin Elliott headed the ball into the path of Maynard, who slammed the ball home from close range after just three minutes.
Derby were eager to claim a quick leveller and Chris Porter forced a save from Robins goalkeeper Dean Gerken minutes later.
The home side had the chance to double their lead after 20 minutes.
Jamal Campbell-Ryce, returning to action following a hamstring injury, crossed the ball in for Louis Carey, whose header in the centre of the Derby goal was punched away by Saul Deeney.
But Campbell-Ryce did not give up and seconds later he had a shot on goal himself which floated past the right-hand goalpost.
Michael Tonge had a chance to score the Rams’ equaliser, but his 18-yard effort was blocked by Robins defender Carey.
Gerken was called into action again 10 minutes before the break. This time Tomasz Cywka tried his luck with a long-range effort, but Gerken made it look easy as he collected the ball from the Derby midfielder.
The Robins got back into the action as the game approached the break as Nicky Hunt blocked Danny Haynes’ effort before Paul Hartley’s corner was quickly cleared by Robbie Savage.
And on the stroke of half-time Gerken made the save of the match as he punched Hunt’s fantastic long-range strike to safety to deny a leveller.
The home side started the second half well and Campbell-Ryce saw his effort go over the crossbar following Hartley’s corner from the right.
McEveley’s corner could have been a danger as it sailed to the near post, but Gary Sawyer was quick to make a clearance to deny the visitors.
And Derby striker Porter almost had the equaliser on the hour with a great effort from the centre of goal, but his left-footed shot was too wide.
County began to dminate as they still searched for a goal and substitute Gary Teale had two chances in quick succession but Bradley Orr was well-placed to clear his first, before Gerken conformably caught his second.
Porter came close again minutes later as he picked up the ball from another Teale cross, but Gerken was ready to save his close-range shot.
Deeney made a great save after 70 minutes to deny Ivan Sproule but Bristol City managed to double their lead nine minutes later.
The 23-year-old striker was brought down by Hunt in the Derby box and Maynard made no mistake as he stepped up to shoot the penalty into the bottom right of the net.
Derby midfielder Stephen Pearson scored a consolation late on as he collected Cywka’s cross and fired home from the centre of goal but they failed to find an equaliser.