Phil Parkinson's side trailed to Brandon Thomas-Asante's close-range finish in the 19th minute at the CBS Arena.
Oliver Rathbone equalised with a composed strike six minutes later.
But Victor Torp netted an 80th-minute free-kick and Ephron Mason-Clark fired home in stoppage time to seal Coventry's 27th win in 45 second-tier matches this term.
Wrexham remain in sixth place, clinging onto the last play-off berth.
They are level on points with seventh-placed Hull, who lost at Charlton on Saturday, and hold the goal difference advantage at +4 over the Tigers' +3.
Derby, in eighth place after beating QPR on Saturday, are also in the play-off hunt, trailing Wrexham and Hull by one point.
Wrexham host Middlesbrough in their final game of the regular season on May 2nd, while Hull are at home to Norwich and Derby meet Sheffield United at Pride Park.
Millwall, Middlesbrough and Southampton are already assured of play-off spots.
Coventry's promotion was confirmed last week and they wrapped up the Championship title by thrashing Portsmouth on Tuesday.
Frank Lampard's team will be back in the Premier League next season after a 25-year absence.
Ipswich hold the second automatic promotion place above Millwall on goal difference.
Kieran McKenna's side will return to the Premier League 12 months after relegation if they take four points from their last two games against Southampton and QPR.