Windass, who missed the victory over Port Vale after picking up five bookings, volleyed home his 11th goal of season to put Bradford on course for back-to-back wins.
However their lead was short-lived and they looked to be heading for defeat when Swans' striker Adebayo Akinfenwa put them ahead after a goalkeeping blunder.
But an edgy encounter erupted after Swansea front man Rory Fallon had a match-winning third goal ruled out for offside with 12 minutes to go.
With three minutes remaining, and most eyes up field, Windass was sent crashing to the ground in the Swansea box sparking furious protests from the home fans and players.
After consulting his assistant referee Eddie Ilderton red-carded Swansea's opening scorer Kristian O'Leary and awarded Bradford a penalty. The decision led to further protests, but eventually Windass held his nerve to fire home the spot-kick and level the scores.
Even then there was still time for Swansea midfielder Marcos Painter to rattle the Bradford woodwork with a superb shot deep into the six minutes of injury time.
After their FA Cup heroics Swansea started on a high and, despite missing a host of players through injury and suspension, they looked the more likely to score.
But it was Bradford who struck first through Windass after midfielder Marc Bridge-Wilkinson picked out on-loan striker David Hibbert with a chipped pass after 23 minutes.
Hibbert laid on the chance for Windass with a neat header and the striker made no mistake as his right-foot volley from18 yards bounced beyond Swans' keeper Willy Gueret.
Bradford midfielder Steven Schumacher then went close twice with a header and 20-yard drive before O'Leary dived in to draw the visitors level.
Bantams' manager Colin Todd will be unhappy with the way Williams' corner found its way into the City six-yard box where O'Leary got the luck of the bounce as Ricketts tried to push the ball away.
The visitors again started brightly after the break, but it took a shocking blunder from Ricketts to gift them a second after 51 minutes.
A sweeping move ended with the impressive Williams delivering another teasing cross from the left, but Ricketts made a real mess of trying to claim the ball on his six-yard line and Akinfenwa was waiting to side-foot the loose ball home from eight yards.
Williams was again the provider when Fallon's effort was ruled out, but Bradford dug deep and will say they deserved a share of the spoils despite the late controversy.