Prolific Constable took his Oxford goal haul to 91 after diverting right-back Damien Batt's angled drive past Plymouth goalkeeper Jake Cole to help his side record just their second win since New Year's Day.
The U's target man could easily have doubled his tally but opted to pass when put through on goal by Sean Rigg's header in the 34th minute.
Argyle left-back Onismor Bhasera cut out Constable's pass, intended for attacking team-mates Josh Parker and Tom Craddock, as the U's broke three-on-one.
Plymouth's on-loan Crystal Palace forward Jason Banton forced former Argyle keeper Luke McCormick to save low down to his left in first-half stoppage time.
And Cole then made a brilliant one-on-one stop to deny Oxford's top scorer Tom Craddock in the 53rd minute following a poor back pass from Plymouth right-back Durrell Berry.
Seconds later, down the opposite flank, Constable's shot spun off the foot of the post.
Oxford skipper Jake Wright was forced to head Reuben Reid's goal-bound side-foot volley off the line following a 63rd minute free-kick as his held on for a vital win.