Michael Hector fired Barnet into a 16th minute lead with a low drive from 25 yards. Oxford eventually recovered from that early blow to control the last 20 minutes of the first half, Oli Johnson twice having shots saved.
Scott Rendell set up Constable for a low shot that Bees keeper Dean Brill turned aside, with some difficulty. The former Wycombe striker then took too long to control Constable's pullback after Johnson got behind the Barnet defence.
But Rendell was in the perfect place to score his first goal for the club, burying a header past Brill from Lee Holmes' superb left-wing cross just before half-time after good work by Peter Leven.
Oxford began the second half strongly as Johnson drove a shot against the inside of the post. And Constable slammed home from close range on 56 minutes after two of his team-mates had efforts blocked in front of goal.
Referee Stuart Attwell awarded Barnet a spot kick when Damian Batt tripped Ricky Holmes just inside the box on 83 minutes. But Clarke brilliantly saved Izale McLeod's effort.
