The hosts were forced to make three changes to their line-up with Gary Monk, Kevin Austin and Leon Knight starting in place of the injured Izzy Iriekpen, Steve Watt and Lee Trundle.
Three times in the opening five minutes they opened up the visitors' defence with lightening raids down the flanks.
Leon Knight had Bantams goalkeeper Donovan Ricketts scrambling out of his goal but Leon Britton was only able to stab the loose ball wide.
In the 12th minute Tom Williams raced down the left before curling a cross into the danger area, but the ball had drifted behind the line, much to the visitors' relief.
Two minutes later Bradford mounted their first attack when Bobby Petta lifted a cross to the far post only for Dean Windass' header to land on top of the net.
Rory Fallon almost claimed his first goal for the club after 15 minutes when he climbed above Bantams skipper David Wetherall, but his crisp header flew over the bar.
Knight was next to have a shooting chance but Ricketts dived low to palm his effort behind for a fruitless corner.
Bradford should have gone ahead in the 34th minute when a Petta cross ran loose inside the home six-yard box. All it needed was a touch from Windass but he lost his balance and failed to tap into an empty net.
The deadlock was broken in the last minute of first-half injury-time. A long ball down the middle was chased by Knight, who turned Wetherall before flashing his shot high into the top corner.
Swansea started the second period brightly with both Knight and Andy Robinson going close before Monk glanced a header wide from a corner.
Out of the blue the Bantams equalised in the 72nd minute when Wetherall rose unchallenged to head home a Marc Bridge-Wilkinson corner.
Kenny Jackett threw on all three substitutes three minutes later in an effort to force the victory the Swans so desperately needed, but the Bradford defence held firm.