Darren Ward headed a 90th minute winner to give Kevin MacDonald the perfect start to his managerial reign.
After falling behind in the first half, two late goals secured three points for Swindon.
The new Town boss made two changes to the starting XI that had lost to Bury in midweek. In came Nathan Thompson and Simon Ferry for Raffa De Vita and Alan Navarro who both dropped to the bench. There was also a return for Joe Devera who took his place amongst the substitutes.
Town almost made a wonderful start – from the kick-off James Collins was denied by Joe Murphy. Seconds later, the home side were in front. Nathan Thompson was robbed on the right and as quick as a flash, Franck Moussa had rifled the ball beyond Foderingham and into the net.
Despite the setback, Swindon proceeded to grab the game by the scruff of the neck: No-one could connect with Andy Williams’ low cross before Murphy was called upon to tip Ferry’s fizzing effort over the bar.
Williams then missed two guilt-edged chances. Firstly he wriggled free of his man and slid his effort wide of goal. He then managed to strike the foot of the post then, as the ball rolled back towards him but his follow-up effort was deflected over the crossbar.
Roberts curled a free-kick tamely at Murphy and the visitors had two clear-cut chances before Town went into the break a goal down.
Swindon began the second half in the same manner with Alan McCormack going close. Wes Foderingham then had to be at his brilliant best to deny Leon Clarke, keeping out his point-blank header with a reflex save. Miller then saw his shot blocked before Flint headed over.
Then came the late drama. After a period of pinball in the Town box, substitute Adam Rooney saw his effort twice blocked before Williams poked home on the goal-line.
Then as the clock ticked over the 90-minute mark, Swindon hit the winner. Luke Rooney’s accurately centred corner was nodded home by Darren Ward who leapt highest to plant the ball beyond Murphy and spark scenes of unbridled joy on the touchline and among the 3,000 travelling supporters.
Swindon Town: Wes Foderingham; Nathan Thompson, Darren Ward, Aden Flint, Jay McEveley; Simon Ferry (Rooney L), Alan McCormack (Capt), Tommy Miller, Gary Roberts; Andy Williams, James Collins (Rooney A)
Substitutes not used: Leigh Bedwell, Joe Devera, Raffaele De Vita, Alan Navarro, Louis Thompson.
Coventry City: Joe Murphy, Carl Dickinson, Nathan Cameron, Carl Baker (Capt), Gary McSheffrey, Steven Jennings, Leon Clarke, Cyrus Christie, Franck Moussa, Aaron Martin, James Bailey
Substitutes: Chris Dunn, Jordan Clarke, Conor Thomas, David Bell, Cody McDonald, John Fleck, Williams Edjenguele
Attendance: 14,280 (3,000 from Swindon)