City striker Craig Farrell, along with midfielder Mark Convery, passed a late fitness test to play and enable boss Billy McEwan to name an unchanged side.
United boss Jimmy Quinn was without left-back Trevor Robinson following his red card in Saturday's defeat at Gravesend.
With no obvious replacement, Quinn was forced to reshuffle the shape of his defence.
Straight from the kick-off Darren Craddock conceded a needless free-kick and as the ball was swung into the box Ritchie Hanlon's header was spilled by keeper Tom Evans, but he recovered to pounce on the loose ball.
City hit back when Nathan Peat's long ball down the channel was missed by Matt Bloomer and Farrell managed to flick an effort goalwards from a tight angle which keeper Paul Crichton pushed round the post.
On a slippy surface City defender Luke Foster twice lost his footing as Marcus Richardson bore down the right flank, but the U's striker was wasteful with his cross.
After those early scares City asserted control and carved out a number of chances. Clayton Donaldson slid a low cross beyond Crichton but Convery was unable to stretch out an connect.
However, Donaldson with his eighth goal of the season put the Minstermen ahead on 22 minutes. Steve Bowey clipped the ball over a square defence and as Crichton raced out of his goal, Donaldson headed over the stranded keeper into the net from 18 yards.
Craddock was replaced by Anthony Lloyd on 27 minutes after a yellow card for a bad foul on Courtney Pitt, before the Cambridge wing-back tested Evans with a well struck long-range effort.
All City's eight wins so far this campaign have been by a solitary goal and once again chances went begging before the break as City sought a two-goal cushion.
Peat intercepted a ball 35 yards out, strode forward and hit a stinging drive from 25 yards which brought a fine flying save from Crichton.
Before the break Farrell twice laid excellent passes into the path of Martyn Woolford who shot at Critchton and then wide, both from 15 yards.
Woolford had three early chances to increase City's lead after the break. First Donaldson picked him out with a cross, but his volley on the run went wide.
Then from another superb Donaldson cross to the far post, Woolford headed straight at the keeper. Woolford then failed to beat Crichton with another header, this time from Farrell's pin-point cross.
Foster twice saved the Minstermen with last-ditch tackles before Farrell shot into the side-netting at the other end. Crichton dived to push Donaldson's cross round his near post and from the resulting corner Michael Morrison headed Donaldson's hooked effort over the bar.
City's failure to convert their chances cost them dear when Ben Sedgemore lashed home an equaliser on 68 minutes when Evans' punched clearance from a corner fell at his feet 12 yards out.
As City hit back it was that man Woolford again missing with a flashing volley from a corner and then yet another header missed the target from Lloyd's pin-point cross.
Substitute Danny Carey-Bertram missed an open goal from six yards out after Evans had parried a long-range shot, but Brady scored the winner three minutes from time direct from a left-wing corner.
There was still time though for Farrell to miss a gilt-edged chance to equalise on 90 minutes when clean through on goal, but again the effort went straight at the keeper.