The striker found the net from the edge of the penalty area within 30 seconds of coming on as a half-time substitute.
It might have been more comfortable for the U's, but Coventry goalkeeper Joe Murphy made outstanding saves to deny Deane Smalley (twice) and Callum O'Dowda.
A victory against a League One side and a first clean sheet of the summer were useful boosts for the U's, but the afternoon was soured by a worrying-looking injury to Jon Meades being stretchered off.
The midfielder was stretchered off with his left leg in a brace and taken straight to hospital for an x-ray.
United's starting line-up included Fulham winger Ryan Williams, who is on trial, while Johnny Mullins came into the side to begin his second spell.
There was no place in the squad for either Dave Kitson or Sean Rigg, who were both rested with one eye on the opening day of the season in a week's time.
It took 15 minutes for the home side to threaten, when Smalley met Scott Davies's corner with a looping header which a back-pedalling Murphy did well to tip over.
The game lacked the same tempo which had been on show for United's 1-1 draw with Birmingham a week earlier.
Both sides knocked the ball around without much penetration and most of the game before the break was played in the middle third.
Mullins was a calming presence, though, picking up where he left off last autumn with a series of quietly effective contributions.
Smalley, United's top scorer in pre-season, again looked the most likely to score and forced an even better save from Murphy nine minutes before the break.
Hunt crossed looking for Callum O'Dowda, but the ball fell to Smalley on his left foot and his powerful effort was brilliantly turned round the post by the Sky Blue goalkeeper.
At the other end it took 40 minutes for the visitors to exert Ryan Clarke.
It came via his namesake Leon, whose curling free-kick was smartly parried away for a corner.
A flat game needed a goal and thankfully it took less than 30 seconds of the second half for one to arrive.
Constable, on for Smalley as one of three half-time substitutions, latched on to a loose ball and drove low into the net left-footed from 18 yards.
Only another superb save from Murphy prevented United from quickly doubling their lead.
Asa Hall threaded a pass into O'Dowda, who jinked past his man and prodded a shot which the goalkeeper did well to keep out one-handed.
The game began to open up and Coventry pressed for an equaliser.
Their efforts were not helped by some woeful shooting, with the ball boys in the car park kept busier than Clarke.
The United keeper was beaten just after the hour mark when Jordan Clarke rose highest at a corner, but Josh Ruffels was in the right pace to clear off the line shortly after coming off the bench.
A good afternoon was spoiled by a worrying-looking ankle injury to Jon Meades midway through the half.
Both sets of supporters stood and clapped as the summer signing was carried off.
It took time for the game to regain its rhythm, but City went on to create their clearest chance of the afternoon.
Hines did well to get the better of Hunt in the area. The winger drew Clarke and squared for Moussa, who blazed over with the net gaping.
United ran the clock down well to secure a victory which will provide a boost heading into the final week of pre-season.
Their one remaining game is at Truro City on Monday, but is unlikely to consist of any first-team players.
Oxford Utd: Clarke, Hunt, Mullins, Wright (Raynes ht), Meades (Newey 73), Davies (Marsh ht), Whing (Ruffels 62), Hall, Williams (Bevans 67), Smalley (Constable ht), O'Dowda. Subs not used: Crocombe, Potter, Rose.
Coventry City: Murphy, Phillips, J Clarke, Willis, Adams, Lobjoit, Fleck, Thomas, Hines, Moussa, L Clarke. Subs not used: Maund, Burge, Garner, Azille-Moore.
Attendance: 1,557 (301 visitors).
Referee: A Coggins.