A Jacobs' penalty gave Northampton the lead but substitute Connell levelled for Swindon midway through the second half.
The visitors were then reduced to ten men but not only survived through to the finish, they inflicted late heartbreak on the hosts with a stoppage time winner that leaves them bottom of League Two.
Aidy Boothroyd will be desperate for the opening of the transfer window and hope that allows him to make the necessary improvements needed to guide the Cobblers away from the relegation zone.
"The players were distraught at the end but we've got a new year and a new season, the second half of this season, to put things right," he said afterwards.
"Players will be coming into the club next week, but for the players that remain at the club, their attitude and their work-rate I can't question."