Seamus McCarten gave Stanley a shock lead but Rovers looked primed to pick up their first win in five games thanks to goals from Tommy Rowe and John Marquis either side of the break.
Stanley claimed a share of the spoils though when substitute Jonny Edwards pounced from close range in the 81st minute.
Rovers, who were strangley hesitant at the back and were guilty of missing some excellent chances, remain three points clear of second-placed Plymouth Argyle but saw the gap to fourth cut to ten points.
Harry Middleton replaced Jordan Houghton in midfield and Alfie May made his first start, replacing Andy Williams in attack. Fit-again Mathieu Baudry had to make do with a place on the bench.
Accrington went ahead with the game's first attempt on goal on four minutes. A quick break saw McCarten lead the charge and run unchallenged from halfway before drilling a shot past Ian Lawlor.
Rovers were unusually hesitant at the back and Billy Kee went close to doubling the visitors' lead when he lashed a shot just over from the edge of the box.
May wasted a good opportunity to level on 14 minutes when John Marquis picked him out in the box but he wanted too much time and saw his effort blocked.
However, it wasn't long before Rovers did equalise. Conor Grant's cross from the right found Rowe whose first effort was saved by Marek Rodak but he poked home the rebound.
Rodak rescued Stanley moments later when he denied Coppinger one-on-one after some lovely Rovers build-up play, before the remainder of the half saw the visitors' spoiling tactics come to the fore.
Rovers made a positive start to the second half as Marquis volleyed wide from Rowe's cross. It proved to be a sighter for the striker, who fired Rovers ahead just before the hour mark.
Coppinger's diagonal ball was flicked on by May into the path of Marquis, who confidently placed his shot past Rodak.
Sandwiched either side of the goal, Lawlor had to be alert to save a McCarten shot from distance and Rommy Boco saw his cross from the right cannon back off the post.
Rovers looked the more likelier to extend their lead and went close through Rowe, Coppinger and Williams, on as a late replacement for May.
But the Keepmoat Stadium was stunned into silence when Stanley grabbed an equaliser on 81 minutes. It came from a free-kick which was headed back across goal and substitute Edwards converted from close range.
Rowe had a glorious chance to win it in the closing stages after being played in by Williams but guides his shot inches wide.
Rovers: Lawlor, Alcock, Butler, Mason, Blair, Middleton, Grant, Coppinger (Baudry 80), Rowe, Marquis, May (Williams 80). Subs not used: Etheridge, Evina, Keegan, Longbottom, McSheffrey.
Accrington: Rodak, Pearson, Rodgers, Hughes, Boco (Shaw 75), Clark, Brown (Edwards 67), Conneely, Clare (Husin 46), McCarten, Kee. Subs not used: Davis, Gornell, Sykes, Chapman.
Referee: Rob Lewis (Shropshire)