Kidlington fought back superbly following the early setback of conceeding from a cooly-taken Josh Ashby penalty.
Home striker Olly Hazell hauled the Colts level before two quick-fire goals from Jacob Haynes and Hazell again put them 3-1 up midway through the second half.
City then came flooding forward in order to try and get something from the game and Ashby grabbed his second late on to setup a tense finale.
Colts defended resolutely though to thwart the visitors and claim the win.
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