Fletcher pounced to score from close range when Rovers goalkeeper Steve Mildenhall spilled a right-wing cross from Lanre Oyebanjo.
But the lead lasted only a couple of minutes before indecisive York defending inside their own box allowed Eliot Richards to poach an equaliser with a low drive.
Carson then lit up the game with a superb dipping volley from 30 yards in the 41st minute to give the visitors the advantage at the break.
Matt Harrold levelled with his first touch after coming on as a 54th-minute substitute, heading in a Lee Brown free-kick and loan signing Chris Beardsley nodded the hosts in front from Michael Smith's 75th-minute cross before Fletcher had the final word.
Carl Baker scored one and played a big part in another as Coventry made the most of stand-in goalkeeper Seb Brown's error to beat AFC Wimbledon 3-1.
Kick-off was delayed until 8pm after a traffic accident held up some of the Dons squad, with the home side forced to replace regular goalkeeper Ross Worner with deputy Brown.
And it was Brown's error that allowed Callum Wilson to equalise Michael Smith's opener before Baker ensured the League One side defeated their League Two opponents.
The Dons should have led five minutes before half-time when Peter Sweeney slipped in Smith only for the striker to fire at goalkeeper Joe Murphy's feet.
Murphy earlier denied Harry Pell and George Porter and the goalkeeper was forced to deny Smith again at the start of the second half.
However, Murphy was finally beaten after 54 minutes when Smith headed in from Porter's cross, but six minutes later the Sky Blues led.
First Brown fumbled Cyrus Christie's cross allowing Wilson to poke the ball over the line and then Baker finished a far-post cross.
And Brown's miserable night continued when Baker's free-kick deflected in off Callum Kennedy in the wall in the 70th minute.
- Sports & Recreation
- Steve Mildenhall