Burton, who netted his first goal for the club in Saturday's draw with Bradford City, stooped to head home a cross from Michael Keane on 70 minutes and knocked the stuffing out of a spirited Vale side.
Vale had the best of the first half, which was notable for a stunning reflex save by home goalkeeper Neil Cutler from a close-range header by Billy Paynter.
Rotherham went ahead against the run of play on 40 minutes when John Mullin fired in a low hard cross and substitute Steve Rowland diverted the ball past his own goalkeeper, Mark Goodlad.
Vale skipper Michael Cummins equalised in first-half stoppage time when his looping header from a Mickey Bell cross beat Cutler and dropped in at the far post.
The Millers stepped up the pace after the interval and midfielder Paul McLaren rattled the bar after being teed up by Michael Keane, and a glancing header by Mullin had Goodlad scrambling.
Burton's goal settled home nerves but Vale again came close to levelling when Cummins hit the bar with a 20-yard free-kick.
Rotherham went all out to kill the tie late on and Keane and Burton went close to adding to their lead.
Vale's problems deepened when substitute Hector Sam limped off injured five minutes from the end, reducing the visitors to ten men.
The Millers sealed their passage into the next round when Jon Otsemobor headed in a cross from Gregor Robertson in the last minute of normal time.
Mullin could have made the scoreline more emphatic when he was played into a great position close to goal, but he ballooned his volley over the bar.