The defender headed in a 94th-minute equaliser after Lee Molyneux's stoppage time free-kick appeared to have won the game for the hosts.
It was a breathless finish, with James Constable cancelling out a wonder goal from James Beattie ten minutes from time.
Raynes' goal gave some comfort to players, staff and supporters, who had begun the day hit by the news former U's winger Mitchell Cole had died, aged just 27.
Jon-Paul Pittman had given United an early lead, but Craig Lindfield sent the sides in all square.
Pittman was one of two visiting players to be forced off through injury, with Andy Whing also succumbing to problems.
Pittman was promoted to United's starting line-up in the only change from the 2-1 win against Northampton Town a week earlier.
The striker replaced the ineligible Jake Forster-Caskey, with Sean Rigg dropping back to left wing.
Rigg provided the best moments of the early exchanges with two direct runs.
The first chance, in the opening minute, was the best and finished with James Constable having a shot saved by Paul Rachubka, who was making his debut.
The ex-Manchester United goalkeeper was signed at short notice on loan from Leeds United after Stanley's only fit senior goalkeeper Ian Dunbavin was injured on Friday.
United had to weather a fierce spell from the hosts around the ten-minute mark.
Padraig Amond rounded Ryan Clarke and had a shot cleared off the line by Michael Raynes.
The U's goalkeeper was forced to make saves from Luke Joyce and Peter Murphy in quick succession. He was beaten by Rommy Boco's header which hit the crossbar shortly afterwards, but the winger was offside.
Stanley were on top, but they were stung by some individual brilliance from Pittman.
Chasing a ball down the left channel, he beat Aristote Nsiala before going past Rachubka and firing past a defender on the line.
It was the second goal in as many games for Pittman, whose pace petrified Nsiala for the rest of the half.
Accrington were still well in the game despite falling behind and they levelled midway through the half.
Boco's ball to the back post found Amond, whose mis-kick fell perfectly for Lindfield to force the ball into the net from close range.
There was little else noteworthy in the game up to the break, as both sides struggled for fluency in possession.
United were hit by the loss of Pittman in stoppage time. The striker went down holding his shoulder and had to be stretchered off.
Cox was substituted for Damian Batt at the break and incredibly United had to make a third change just four minutes after the restart.
Whing, who had moved from right back to midfield to accommodate Batt, went in for an innocuous tackle but ended up writhing in agony. For the second time in a handful of minutes the stretcher was needed to carry off a U's player.
Accrington sensed a moment of weakness and Clarke had to save from Boco and Beattie.
But after a brief spell United adjusted to their changes and came back into the game.
Just before the hour Rigg had a shot tipped over and from the corner Liam Davis and Tony Capaldi had efforts blocked by last-ditch challenges.
Rigg, who had been pushed up front after Pittman's exit, had another good sight of goal on 75 minutes, forcing a smart save from Rachubka.
United looked the most likely to score, but they were stunned ten minutes from time.
A long ball forward was only half cleared by Batt and in a flash Beattie hit a half volley from 25 yards which flew into the top corner via the post.
United were staring at a cup exit, but roused themselves to draw level six minutes later.
Alfie Potter slid a pass into Constable, who confidently finished for his fifth goal of the season.
Accrington thought they had won the game when Molyneux curled in a set piece, but Raynes provided one final twist in the tail to keep United in the cup.
Oxford Utd: Clarke, Whing (Chapman 49), Raynes, Wright, Capaldi, Potter, Cox (Batt ht), Leven, Rigg, Pittman (Davis 45), Constable.
Unused subs: Brown, Heslop, Worley, Marsh.
Accrington: Rachubka, Winnard, Nsiala, Murphy, Liddle, Lindfield (Mingoia 90), Joyce, Miller, Boco, Amond (Molyneux 68), Beattie (Gray 85).
Unused subs: Dawber, Eckersley, Barnett, Hatfield.
Referee: Andy Woolmer.
Attendance: 1,195 (248 visitors).