The home side, who were searching for their first League One win at home in more than two months, took the lead from the penalty spot in the 32nd minute after Carl Baker brought down the jinking Filipe Morais.
Midfielder Lucas Akins fired home at the second attempt after his first spot-kick was disallowed for encroachment.
Coventry pushed forward in search of an equaliser after the break and with Boro dropping deeper, defender Richard Wood reacted quickest to a blocked John Fleck shot to tap in and get the Sky Blues back on level terms after 79 minutes.
Two minutes after Stevenage had a Mark Roberts header cleared off the line, Baker's curler from the edge of the box gave Coventry the lead in the 89th minute.
And the points were secured a minute later as David McGoldrick ran from his own half before chipping goalkeeper Steve Arnold from 25 yards out to bag his 18th goal of the season.