In his third start since making his comeback from a serious knee injury, Maynard struck his third goal to set the Robins on their way after 15 minutes.
Lewin Nyatanga poked home David Clarkson's free-kick nine minutes later to give Keith Millen's side a two-goal cushion and Clarkson put the visitors firmly in control a minute before half-time from the penalty spot.
Coventry were given a glimmer of hope when Lukas Jutkiewicz pulled a goal back eight minutes into the second half, but that hope was shortlived as Marlon King was sent off on the hour mark.
Marvin Elliott completed the scoring after 82 minutes to give Bristol City a third straight win and move them 10 points clear of the drop zone.
Conversely, Coventry have won only one of their last 14 league matches and have slipped to 19th place in the standings, just six points above the bottom three.
Both sides made one enforced change following encouraging performances last weekend.
Martin Cranie served a one-match suspension after his red card in the draw at midlands rivals Leicester, so Sammy Clingan came back into Coventry's midfield.
On-loan West Ham defender Jordan Spence made his Bristol City debut. The youngster was signed on loan this week as cover for Louis Carey, who faces a lengthy spell out after suffering a fractured skull in the home victory over Scunthorpe. Spence replaced the City captain at right-back.
Coventry made a positive start so it was against the run of play when Maynard latched on to Clarkson's flick and confidently beat Keiren Westwood from 12 yards.
Clarkson was the provider again soon afterwards, swinging in a free-kick from the left for Nyatanga to touch home.
Weak efforts from Jutkiewicz and Aron Gunnarsson either side of the half-hour mark were all Coventry could muster and they were three behind heading into the break.
Nathan Cameron fouled Albert Adomah and Clarkson made no mistake from the spot.
To make matters worse for the Sky Blues Westwood had to be replaced at half-time due to injury, meaning 20-year-old Danny Ireland made his Football League debut for the club.
However, Aidy Boothroyd's side responded in adversity and pulled a goal back after 53 minutes.
Clingan's free-kick was headed back into the danger area by Richard Keogh and Jutkiewicz scored for the first time in 17 games from close range.
Ireland kept his side in the game immediately afterwards with a fingertip save to divert Clarkson's shot past the post.
But King ruined Coventry's optimism on the hour when he was sent off for violent conduct.
Bristol City dominated thereafter with Maynard and Martyn Woolford going close before Elliott converted Adomah's right-wing cross from 12 yards out with eight minutes remaining.
Ireland excelled again late on, denying the visitors a fifth goal with a brilliant block to thwart substitute Brett Pitman.