A fine strike from Emily Roberts and a late own goal from Ciara Watling saw Palace fall to defeat in the north-east.
Having won their first ever game in the Women’s Championship the previous week against the London Bees, Dean Davenport’s side were optimistic going into the game but ultimately made the long journey home without anything to show for their efforts.
The hosts started well with Rachel Lee curling an early effort over and Roberts flashed an effort just wide of the far post before Cottam firing over after a lightning quick counter-attack.
Palace had their moments, but failed to seriously test Hannah Reid who was merely a spectator for the most part - although she did have to remain alert to hold a curling effort from Kallie Balfour.
Durham, though, were still pouring forward in numbers and Beth Hepple and Roberts came agonisingly close to open the scoring as frustrations in front of goal continued when Ness and Lee both squandered late opportunities to take the lead as the game remained goalless at the break.
But it took the hosts just over 10 minutes of the second half to find that elusive breakthrough as Roberts curled into the bottom corner from 12 yards out.
The visitors saw more of the ball as they sought an equaliser but failed to seriously trouble Reid as the Wildcats' back four remained solid, and Durham doubled their lead with just five minutes remaining when a mix-up between stopper Shannel Salgado and Watling saw the latter head into her own goal.
Reid made a late save from a Jade Keogh free-kick, but the Wildcats held on to claim the victory.