An early mistake gave away a goal to put the Bantams in trouble and they had another big problem when they were down to ten men for most of the second half.
Big Dennis Lawrence, back from World Cup duty with Trinidad and Tobago, started and finished a four-man move, beating the offside trap to jab his shot past a surprised Paul Henderson only 84 seconds into the game.
From then on Bradford were chasing the game and although they created several chances they couldn't catch up. Strikers Dele Adebola and Dean Windass were closely marked but both had openings that came to nothing.
Windass saw a volley charged down by the defence and twice missed from free-kicks, placing the first tamely wide and the other high over the crossbar.
Windass thought he had scored eight minutes before half-time when he stooped forward to head Darren Holloway's right-wing pass into the net but the whistle had gone for offside.
Lee Crooks was booked early on and when he tangled with Lawrence in the second half, referee Mark Warren sent him off for a second offence. Adebola was also out of luck when his volley from Wayne Jacobs' cross hit startled goalkeeper Andy Dibble in the chest and bounced up over the crossbar.
Dibble had escaped narrowly earlier when he picked the ball up outside his area, receiving only a yellow card.
Wrexham did enough to keep the Bantams on edge and Chris Armstrong was not far off with a header that flashed past a post and another that sent Henderson back-pedalling to clutch it under the bar.
Bradford also had chances when Tom Kearney shot narrowly off target, Mark Bower headed wide and Steven Schumacher slipped when unmarked with the ball at his feet only ten yards out.
Then in the dying seconds substitute Michael Symes was in at the near post and fluffed his shot, giving an easy save for Matt Baker who made his league debut, replacing Dibble for the second half.