The 28-year-old, who was capped six times by his country, struck twice in the opening 11 minutes to give the Millers their first win in normal time in nine games.
Rotherham had claimed a Johnstone's Paint Trophy success on penalties in midweek.
Burchill opened the scoring in the fourth minute when he converted from the spot after being brought down by Colin Murdock.
The ex-Celtic man made it two seven minutes later when he converted Dale Tonge's delivery from close range.
Andy Proctor gave Stanley hope when he fired home on 33 minutes. The midfielder converted into the bottom right corner from inside the box.
The hosts huffed and puffed in the second half but have rarely looked like imposing themselves on a home game for a number of seasons.
Rotherham always looked more likely to extend their advantage with star man Burchill looking to grab himself an early Christmas present in the shape of the matchball for a hat-trick.
But in the end it was Drewe Broughton who extinguished any of those home hopes with three minutes left when he slotted home Rotherham's third from close range.