It was the hosts who took the lead just before half an hour had been played, when Tunde Owolabi nipped in ahead of the Hoops backline and fired home Rick Smith’s cross from point-blank range.
City then withstood a bit of pressure before levelling the game three minutes into first half stoppage time.
A ball found its way through to Corie Andrews, who kept his composure to take the ball round Mateusz Hewelt and slot in from a tight angle.
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The home side threatened from set pieces during the second period, and the Hoops needed Kai McKenzie-Lyle to make an important save at the feet of substitute Mo Doro to keep the scoreline level.
City nudged in front shortly after the hour mark when a lovely ball from Andre Burley found Jacob Roddy, who teed up Zac McEachran for a shot.
The effort was spilt by Hewelt into the path of Tom Scott, who made no mistake with his finish.
Radcliffe then attacked in numbers in search of an equaliser, but were caught out when a lapse in concentration from Smith allowed Josh Parker to play in McEachran, who drove forward and slotted home when one-on-one with the goalkeeper.
Three points for the Hoops lifts the side above the bottom four, three points clear of Radcliffe, plus Rushall Olympic, Marine and Needham Market, who all have 10 points.