The visitors got off to a dream start after six minutes when Colin Murdock was judged to have pushed Burchill in the area and the striker dusted himself down to send keeper Ian Dunbavin the wrong way from the spot.
The Millers then added a second four minutes later when Dale Tonge’s low cross was stabbed home for his second of the game by Burchill from four yards out.
However, Stanley got back into the game shortly after the half hour mark when Chris King’s cross was met at the far post by Andy Procter who rifled the ball into the roof of the net for his first goal of the season.
Accrington piled on the pressure after the break as they looked for the equaliser with Jamie Clarke, John Mullin and Jimmy Ryan all going close.
But Rotherham wrapped it up two minutes from time when Jamie Green’s cross was met by substitute Drewe Broughton who fired past Dunbavin from close range.
The home side have now only registered one win in their last ten games.