The visitors had looked to be heading to a defeat as Coventry had the best of the first half, but Maynard's stunning strike was worthy of a point.
Coventry City manager Chris Coleman took the bold step of axing star midfielder Aron Gunnarsson from the starting line-up after a string of below-par performances from the young Iceland international.
But in the first-half fortune certainly favoured the brave as Coventry, after a few scares, charged into a lead.
Bristol's on-loan striker Alvaro Saborio started well and could well have put them in front in the early stages.
After than initial storm, Coventry began to find their feet. Clinton Morrison was the only player to react when Dean Gerken dropped a soft shot from Isaac Osbourne, but the keeper did well at the second attempt.
Leon Best, who had clearly got the bit between his teeth, did well in the 14th minute to reach a free-kick from Sammy Clingan only for his flick to just miss the on-rushing Morrison.
But four minutes later he got it just right when the returning Marcus Hall curled in a super cross, which allowed Best to emerge from a line of defenders and head home.
Gerken and a linesman then kept Bristol in touch. Firstly the keeper did well to tip a save round the post from Ben Tuner.
Then in the 24th minute, Gerken could not hold onto another header from Best, allowing Morrison to swoop but the former Crystal Palace man had come from an offside position.
At the other end, Saborio was denied by an excellent stop from Keiren Westwood.
Whatever happened at the interval in the Bristol City dressing room clearly had an effect.
They pulled level within three minutes of the restart and really should have taken the lead shortly afterwards.
There was certainly no denying the quality of their equaliser. Saborio chested the ball back well for Maynard and the former Crewe striker let fly with a thumping strike from 25 yards which gave Westwood no chance.