Having lost narrowly at Carmarthen on Saturday, they dominated at Llandarcy Park but were unable to breach a massed defence.
Keeper Paul Smith was a virtual spectator for most of the match, but was let down by defenders Michael Johnson and Marti
n Beattie on 25 minutes as Matthew Rhys took full advantage of their confusion.
That was the signal for the hosts to put everyone behind the ball, while the situation wasn't helped by an over-zealous Dean John, who seemed intimidated by the presence of referees supremo Roger Gifford in the stand.
But for all Bangor's pressure, keeper Craig Morris was not worked enough, although he denied Ashley Stott in the first period, while Les Davies chipped the custodian on the stroke of half time only for the ball to hit the bar.
The siege continued in the second half with Mike Walsh skimming the bar with a cross, while Davies had an effort pushed over by Morris.
Davies and sub Karl Noon failed to hit the target from good positions, while Rees failed to capitalise in a rare break for the hosts.
Morris excelled with saves from Davies and Walsh in the closing stages, but it needed the legs of Smith to spare City's blushes when sub Gavin Jones latched onto a long ball from the home keeper at the death.
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