Stanley were left waiting for their first league win of the season but managed to extend their best run so far on Saturday.
Given the Reds had previously suffered eight straight defeats, a run of three draws from four games marks progress for James Beattie's side, who remained bottom of League Two.
Although Stanley did create chances against an Oxford side that came into the game in third place, the Reds were indebted to on-loan keeper Marcus Bettinelli, who made some fine saves in the second half.
Rob Atkinson and James Caton were injured while Kayode Odejayi began a three-game ban and Michael Richardson dropped to the bench.
In came Dean Winnard and fit-again striker Danny Webber. James Gray and Kal Naismith also returned and the two combined early on, Gray chasing a ball down the right and playing it back for Naismith, who fired wide of the far post from just outside the corner of the box.
Oxford replied with Danny Rose hitting a 25-yard effort over the bar before the midfielder curled a free-kick just over and against the stanchion behind the Accy goal.
Naismith beat left-back Tom Newey again in the 29th minute to make his way into the Oxford box and visiting keeper Ryan Clarke did well to stick out his left leg and save Naismith's shot.
David Hunt then delivered an outswinging free-kick from the Oxford right and although James Constable's header was heading inside the post it lacked power, allowing Bettinelli to get across his goal and turn it wide.
Just after the break, Bettinelli raced out of his area to make a well-timed tackle on Constable while Stanley midfielder Piero Mingoia saw his shot deflected wide for a corner, from which Naismith fired over.
Oxford then began to take charge, with Sean Rigg cutting in from the left to fire into the side-netting on 59 minutes.
Former Premier League striker Dave Kitson headed wide from Newey's cross before Hunt's long throw found Michael Raynes, although the centre-half's header was too close to Bettinelli.
The Stanley keeper then denied Rigg low to his right while Constable fired wide and from another long throw he made his most important save.
Hunt again found the head of Raynes, only for Bettinelli to make a superb one-handed save low to his left, with Ryan Willaiams turning the rebound into the side-netting.
The Reds regrouped and pushed for a winner, with Clarke relieved to see Hunt clear it off the line after the Oxford keeper fumbled a cross.
Clarke then dealt with Lee Naylor's stoppage-time free-kick although a Stanley win would have been harsh on promotion-chasing Oxford.
STANLEY : Bettinelli 9, Hunt 7, Winnard 8, Aldred 8, Naylor 7, Naismith 8, Murphy 7, Joyce 7, Mingoia 7 (Hatfield 83), Gray 7 (McCartan 88), Webber 7 (Richardson 71). Subs: Carver, Miller, Windass, Dunbavin.