Doncaster trailed 2-1 at half-time.
They were behind on eight minutes, Rommy Boco netting, before Tommy Rowe equalised for Rovers.
But Accrington edged back in front on 29 minutes when Matty Pearson found the net.
Rovers were back level in the 82nd minute when Andy Williams - who earlier missed a penalty - pounced.
But in stoppage time, Sean McConville pounced to hand all the points back to the hosts.
“It’s about results, it’s a results-based business as well all know,” boss Darren Ferguson said.
“It’s a bad start for us. A bad result. “I think what cost us the game is lapses in concentration.
“It was three really soft goals that we gave away and that has cost us the game, clearly.
“The message to them was make sure we don’t concede, that we don’t lose the game.”