The Grecians charged into a 3-0 lead inside the opening 24 minutes and also missed a penalty early in the second half after Stanley had got themselves back into the game.
Striker Tom Nichols opened the scoring in the sixth minute when he got the ball to the left of midfield and saw goalkeeper Marcus Bettinelli off his line before launching a neat chip over the stopper from 20 yards out.
It was 2-0 after 11 minutes with Alan Gow's quick pass catching the home defence flat-footed and Nichols fired a low, hard shot into the bottom of the net from inside the area.
The third goal was not long in coming. Experienced forward Gow turned and shot inside the box with Bettinelli unable to keep it out at the near post with 24 minutes on the clock.
Stanley got themselves back in it in the 38th minute when Kal Naismith's right-wing cross was headed in by Peter Murphy from close range.
And it was Naismith who made it 3-2 five minutes after half-time as he leapt highest to head home Lee Naylor's cross.
Naylor was then judged to have brought down Nichols in the box and Gow stepped up to take the penalty but his kick was pushed away by Bettinelli.
Grecians keeper Artur Krysiak pulled off two superb saves to keep out efforts from James Gray and Piero Mingoia to deny Accrington an equaliser.