Oli Johnson had opened the scoring in his first start for the U's, before Dagenham equalised through skipper Mark Arber in the second half ahead of Constable's winner.
Johnson, released by Norwich last month, glanced in a deft near-post header from Liam Davis' left-wing cross six minutes into the second half to break the deadlock.
Constable had to clear off the line before Oxford missed a great chance to double their lead when new signing Scott Rendell raced through on goal. But a poor touch allowed Chris Lewington to make a smothering save.
Centre-half Arber levelled in the 72nd minute, planting a header across goal and past keeper Ryan Clarke from Medy Elito's well-flighted cross. Oxford had failed to learn their lesson from two minutes earlier when Elito's centre evaded everyone, only for Jon Nurse to head straight at Clarke.
Neither team had played since January and it showed in a dire first half which was 25 minutes old before a proper chance was created. But the match burst into life in the second half, and Constable settled it with his 10th goal of the season late on.