Miles was shown a straight red for his lunge at Steven Schumacher and with Bradford wingers Jermaine Johnson and Lee Holmes in thrilling form, the visitors succumbed to two goals in the space of four second-half minutes.
Vale keeper Mark Goodlad had already pulled off a couple of stunning saves, but he could do nothing to stop midfielder Marc Bridge-Wilkinson sliding in to force Holmes' pin-point cross home on the hour.
City keeper Donovan Ricketts also made some fine stops and soon after Bridge-Wilkinson's opener he brilliantly tipped away a Danny Whitaker drive.
A minute later Holmes was again teasing his marker to put over another left-wing cross and, when the ball rebounded back off the far post, City striker David Graham was on the spot to stab it home from close range.
Inspired by Johnson's thrilling display on the right wing, City fully deserved their lead and but for Goodlad the victory would have been more impressive.
The Vale keeper denied Bridge-Wilkinson with a stunning tip over after just four minutes and midway through the half he blocked bravely at the feet of Holmes.
Vale hot-shot Leon Constantine wasted a great opening after 13 minutes when he shot wildly despite being clean through on goal and when Miles needlessly clattered into Schumacher they faced an uphill battle to keep Bradford at bay.
Just before the break Ricketts pushed away a vicious drive from Akpo Sodje and in the last attack of the half the Vale frontman headed over the bar.
But with Johnson getting into his stride after the interval, City always looked the more likely to score and Goodlad's heroics could only keep them out for so long.
As 10-man Vale tired in the closing stages, Goodlad kicked away another Johnson strike and Bridge-Wilkinson saw his low drive come back off the inside of a post.