Lucas Akins put Boro in front from the penalty spot but three goals in the final 11 minutes saw the Sky Blues triumph 3-1 and climb into the top half of the League One table.
Richard Wood reacted quickest to a blocked John Fleck shot to tap in and get the visitors back on level terms after 79 minutes.
Two minutes after Stevenage had a Mark Roberts header cleared off the line, Carl 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 for his 18th goal of the campaign.
"We were good all day and good value for it in the end," said Robins. "We had five or six chances in the first half and didn't take any and you start to think something might go against you.
"The goal from Wood gives us that little bit of a lift. The supporters from then have played a massive part in keeping us going, and keeping us going forward.
"McGoldrick was tremendous. He's seen the goalkeeper is off the line. It's all right seeing it but it's the execution as well. It's rounded off a good day for us in the end.
"Make no mistake about it, we're a big club that is punching below our weight. The supporters have started to believe a little bit and we've got to rise to that challenge because they will let us know if we don't.
"From my point of view, it's about managing the expectations but it's also good to be involved and watch at times. We've sent the supporters home really happy."