Martin Allen's men bounced back from their defeat against Colchester with a storming start which saw then score twice in the first half-hour, but City refused to throw in the towel and they were rewarded with a share of the spoils thanks to Bridge-Wilkinson's double.
With City already two down, Bridge-Wilkinson had raised hopes of a comeback when he hit a superb 25-yard free-kick which beat keeper Stuart Nelson all ends up.
But the Bantams looked to be heading for another depressing home defeat until their late rally.
Colin Todd's men were caught out after just four minutes when Brentford striker Dudley Campbell took full advantage of a fatal hesitation on the edge of the City box to fire the visitors ahead with low 20-yard shot.
Soon after Bradford keeper Donovan Ricketts spread himself well to deny the unmarked Campbell with his legs, but there was no let-off after 28 minutes when the tall Jamaican international was harshly adjudged to have up-ended the Brentford dangerman.
Campbell calmly sent the Bantams keeper the wrong way with his spot-kick, but Todd's men were thrown a lifeline by Bridge-Wilkinson's perfectly executed right-foot strike which arrowed into the top corner.
The goal, just before the break, was a much-needed boost and in City's first attack after the interval they were unlucky not to draw level.
Striker Dean Windass saw his glancing header come back off the woodwork and when winger Ben Muirhead fired in the rebound, Brentford centre-half Michael Turner booted the ball off the line.
Just before the hour mark, City's poor defending came back to haunt them when Brentford midfielder Ed Hutchinson was allowed a free-header at a corner and centre-half Sam Sodje dived in unmarked at the back post to restore their two-goal lead.
But City showed commendable fighting-spirit to claw their way back and when skipper David Wetherall hooked in a loose ball 12 minutes from time an unlikely comeback looked a possibility.
With Brentford desperately clinging on, City surged forward and it was after midfielder Tom Kearney had forced his way into the box that substitute Steve Claridge was bundled over by Lloyd Osuwu.
Referee Darren Drysdale again pointed quickly to the spot despite Brentford's protests and Bridge-Wilkinson held his nerve to send Nelson the wrong way with his penalty.