A good run of form for Aidy Boothroyd's side was brought to an end by the U's with the home side opening the scoring through Constable on 15 minutes.
Clive Platt levelled with just under 15 minutes remaining but Boothroyd will have been disappointed to see his side concede the winner to Pittman just a short while later.
Town now lie in 10th place, 2 points behind the play-off places.
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Unhappy with some of the decisions that went against his side, Boothroyd said afterwards: "We weathered a storm and got back in the game; you know against a side like Oxford that the crowd can either work for them or against them and they were just starting to turn after we scored.
“Perhaps we should have got the ball in the box a little bit more because when we did that we seemed to get chances."