Basingstoke took the lead through Taff Richardson after 26 minutes, only for United to hit back thanks to cool finishing by Corey Browne and Stuart Beevor heading in a corner to make it 2-1.
Cliff Hercules had a header cleared off the line a minute into the second period but, apart from Beevor hitting the bar, it was mostly Town who dominated.
Richardson bagged his second goal, and Hurdle beat the outstanding Gallagher to put Stoke 3-2 in front. Last season's top scorer Mancey capitalised on more poor defending to head home a fourth, and United's Ian King summed up the night deep into injury time by blasting a penalty well wide.