Improving Stockport made it three games unbeaten with a hard-earned 1-0 win at manager Paul Simpson's old club Oxford.
George Donnelly got the winner in the 41st minute with his fifth goal of the season.
The on-loan Plymouth striker met Adam Griffin's beautifully-struck left-wing cross with an unstoppable header from six yards out. But Oxford really should have been in front by then.
Matt Green, who constantly had the beating of Griffin down the right, took the ball around keeper Matt Glennon after charging through on goal, only for Danny Swailes and Mansour Assoumani to get back and make a goal-saving challenge.
In the second half, Oxford laid siege to the Stockport goal at times, and Constable headed against the foot of the post when it seemed easier to score.
Glennon saved a swerving shot from sub Alfie Potter, and although Asa Hall headed home the rebound, he was offside and County held out.