Coventry were the stronger side for much of the match - but Jordi Gomez scored a quality set-piece from 25 yards to give them a 67th-minute lead.
But just as the Sky Blues looked to be heading for their third home defeat on the bounce, Fox stepped up to score a matching effort with just three minutes to go.
It was a fair result for Coventry who had every reason to feel that luck had deserted them.
Referee Mark Haywood had denied them a penalty, ruled out two goals and awarded a dubious foul for the all-important Swansea free-kick. The home side also lost right-back Stephen Wright to injury in the first half - only for his replacement, Isaac Osbourne, to hobble off minutes later.
The penalty appeal came in the sixth minute. The ball fell in front of Freddy Eastwood with an open goal to aim at but, as he tried to shoot, Angel Rangel pulled him to the ground - but no foul was given.
A minute later, team-mate Clinton Morrison had the ball in the net when Dorus de Vries parried an Elliott Ward shot to the striker - only for the linesman to rule the goal out for offside.
The home side continued the pressure with Michael Mifsud going close and Robbie Simpson forcing a save from de Vries with a 25-yard drive.
Swansea had to wait until the 59th minute to register their first shot on target - but after Guillaume Beuzelin was penalised for a seemingly fair tackle shortly after, Gomez showed his class to put the visitors ahead.
Coventry continued to push forward and Morrison thought he had scored again in the 77th minute - only to be met with another offside flag.
But with the game nearing stoppage time, Fox stepped up to strike his effort into the top corner to give Coventry a valuable point.