Manager Colin Todd sent out three of his young guns in a game which amounted to little more than a mid-table scrap and there were few moments of excitement in a low-key first half until Bower stooped to claim his third goal of the season a minute before the break.
City had the best of the early opportunities, but leading marksman Dean Windass saw his goal-bound strike well blocked by Vale defender Michael Walsh.
The visitors lost midfielder Danny Sonner after just 18 minutes through injury and they failed to muster a threat on the City goal until Sam Togwell's snap-shot just eluded Chris Birchall on the half-hour mark.
Bantams keeper Donovan Ricketts then made two smart saves in quick succession to deny Birchall and Togwell before Bower broke the deadlock in disputed fashion.
Referee Gary Lewis ruled that Vale keeper Mark Goodlad had carried the ball over the deadball line as he tried to collect Joe Colbeck's wayward cross and the young City winger was soon able to make amends with the resulting corner.
As Windass was sent sprawling in the box, Colbeck's right-wing cross eluded the visiting defenders and Bower stole in to head home from 10 yards out.
The goal was just the boost City needed and they made a bright start after the break but were unable to find the killer touch.
Midfielder Lewis Emanuel saw his well-struck shot fly just wide of the mark after 53 minutes and Vale substitute Tony Dinning got a vital touch to deny Marc Bridge-Wilkinson a free header six yards out.
Ricketts then produced another excellent tip over to keep out a Jeff Smith drive but the City keeper could look on as substitute striker Hector Sam wasted Vale's best opening in the last minute of normal time.
Togwell's cross from the left caught City's backline all at sea, but Hector somehow headed wide despite being unmarked just eight yards out.