Two goals in a minute just before the break from Jess and Blake gave The Bantams the advantage, Blake made it three from the penalty spot, and although McVeigh gave the home side some hope, a last-gasp strike from substitute Bower shattered Norwich's unbeaten home record.
Both managers made one change to their starting line-ups with Darren Kenton coming in for the injured Adam Drury in the Canaries XI, whilst Jim Jefferies handed Peter Atherton his first start of the season in place of Robert Molenaar.
The visitors made a lively start as they attempted to win just their second away league game of the season, with Matthew Etherington in particular looking dangerous, but the first chance went Norwich's way when Aidan Davison parried Iwan Roberts' header.
The best early Bantams effort fell to an unmarked Gary Locke, but he sliced his chance wide from the penalty spot.
Norwich hit back and should have taken the lead when Roberts, in acres of space, shot straight at Davison and the Bantams keeper was again the hero when he made a great save from Marc Libbra.
Bradford turned the game on its head just before the break, however, with two goals in a minute.
Jess headed in Andy Myers's left-wing cross for his 10th of the season and moments later Blake found himself in space inside the area and gave Robert Green no chance.
Ten minutes into the second half it should have been 3-0 to Bradford but Green dealt with Etherington's header, whilst at the other end Roberts missed two more chances to hand his side a lifeline. Davison made a superb one-handed save from his header before the striker fired wide after being sent clean through.
And Roberts was made to pay for his wastefulness in the 64th minute when Blake was adjudged to have been tripped by Green, the Bantams forward stroking home his second of the game from the penalty spot. His celebrations, directed straight at Norwich fans in the Barclay, however only served to incite the Norwich crowd to make for an exciting finish.
McVeigh finally gave Norwich fans something to celebrate three minutes later when he latched onto substitute Alex Notman's clever pass and finished past Davison.
The goal sparked an intense spell of Canaries pressure, with man-of-the-match Davison thwarting McVeigh and Libbra, before a Bradford breakaway saw Bower wrap things up in the final minute to deny Nigel Worthington the chance of topping the Division One table, again.
