Referee Carl Boyeson had incurred the wrath of the home fans after his first-half dismissal of crowd favourite Dean Windass for foul and abusive language, but that was nothing compared to the anger generated by his decision to penalise Nicky Summerbee for a trip on Wrexham's Stephen Roberts.
Despite the home protests, which continued after the final whistle, Armstrong kept his head to drive home the left-foot spot-kick and earn a crucial point in their bid to pull away from the relegation zone.
It was a dramatic climax to a contest which began in low-key fashion and it took Windass' departure - City's third sending off in six games - to bring the game to life.
The experienced frontman should have known better when he rounded on the referee in the 27th minute after he had penalised him for a foul on Wrexham skipper Darren Ferguson.
The official had no hesitation in showing Windass the red card and the striker was so aggrieved with the dismissal that he tried to confront Mr Boyeson again before being ushered away by Wrexham defender Dennis Lawrence and a member of the City backroom staff.
Prior to Windass' departure the visitors had looked the more composed with Ferguson being given plenty of room in midfield.
A Ferguson corner was met well by Lawrence after 14 minutes but his powerful header was kicked off the line by Bradford full-back Lewis Emanuel.
Five minutes after the break Lawrence's clever through ball found Wrexham striker Chris Llewellyn in space, but his 14-yard shot was too close to keeper Paul Henderson and the gilt-edge chance went begging.
It proved a let-off for City and they made the visitors pay nine minutes later when a well-worked move sent Summerbee racing into the box and when his chip beat keeper Andy Dibble, Bradford midfielder Steven Schumacher slid in to force the ball over the line.
The goal boosted City's confidence and they made light of Windass' absence without really theatening to score the vital second.
Armstrong headed a great chance wide after 63 minutes but it looked to be City's day until Summerbee sent Roberts tumbling in the box.
The final whistle was the cue for more protests from some of the Bradford players and staff and stewards had to restrain one fan who sprinted onto the pitch and got within 10 yards of Mr Boyeson.