Roy Walker's Ballymena are still bottom of the Premiership
James Slater scored a last-minute goal in a match which was overshadowed by a serious injury to Albert Watson.
Play was held up for 17 minutes early in the game as Watson was treated on the pitch for a badly gashed leg.
Aidan Watson was sent-off for a second yellow card challenge on Jamie Tomelty on 49 minutes and Ryan Mullan swerved a shot past Alan Blayney on 57 minutes.
United levelled after 82 when Neil Teggart followed up to score after Dwayne Nelson saved from Kevin Kelbie.
Adam McMinn headed on a Dwayne McManus corner in the final minute and Slater got a touch to secure a vital three points for Swifts.
After Watson was stretchered off in the first half, Mullan had a shot blocked and Blayney denied Mark McAllister.
Ballymena's misery was compounded as manager Roy Walker was sent to the stand in the closing stages.
It was feared that Watson had fractured his leg but hospital tests revealed that his left leg was not fractured, just badly gashed.
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