Darlington were happy to be back on home turf after a three-week absence and they went close to opening the scoring after five minutes following a mistake from Vale keeper Joe Anyon.
Striker Liam Hatch almost capitalised when the keeper dropped the ball, but a quick recovery saved the day for the visitors.
The closest that Vale came was a shot from the edge of the area by Simon Richman that was deflected away from the danger area by Ricky Ravenhill in the 15th minute.
The home side were forced into a change after 22 minutes when David Poole was replaced by Adam Griffin following a strong challenge on him by Louis Dodds.
The game burst into life when a Luke Rodgers free-kick from the edge of the Darlington area looped over home keeper Simon Brown and into the far corner of the net to put the visitors into the lead in the 29th minute.
Vale's joy was shortlived as, on 38 minutes, Darlington equalised through Burgmeier, who saw his low free-kick sneak in at the near post.
Seconds before the break, Vale defender Luke Prosser was booked after a cynical tackle on Ravenhill stopped the midfielder in his tracks.
Burgmeier went close in the 50th minute with a snapshot from the edge of the area as Darlington went close to taking the lead.
But the visitors stormed back to dominate the half. Goalscorer Rodgers sent over a low cross on the hour mark that the Darlington defence managed to scramble clear.
In the 71st minute a left wing cross from Robert Taylor was headed wide by captain Marc Richards and then Darlington keeper Brown saved his side as he clawed away another effort from the lively Rodgers.
After surviving the Vale onslaught, Darlington took the lead somewhat against the run of play in the 77th minute.
Substitute Adam Griffin's corner was met by Hatch and the on-loan frontman headed in what proved to be the winning goal on his home debut.