On a pitch that resembled a ploughed field, the visitors adapted well in a tough and at times brutal match against physical opponents.
Ponte' created all the best chances against a suspect defence.
Pacey strike pair Danny Maw and Irvine Machama both went close as Hartshead struggled with the ball over the top.
Jason Kitchen went close midway through the half but the hosts' keeper was in inspired form, denying Ponte' on several occasions.
The home side relied on a long ball to their lone front man but he was ruthlessly shackled by Social centre-back Danny O'Hearne, who was in dominant mood.
Ponte' had to wait until the second half to make their superiority tell, Kitchen blasting home after Maw's shot was only parried by the keeper.
Best for the winners were Shaun Palfrey, Danny Stone, Jason Dobson and game-star O'Hearne. James Jenkins and Stephen Westwood impressed off the bench in the second period.
Kellingley Welfare, who had no game at the weekend, return to action with a trip to Old Centralians on Saturday. Players to meet at the club before 12.15pm.