Shanade Hopcroft had a great opportunity to put Palace back into the lead, but her volley was saved brilliantly by Brosnan in the Everton net.
It ended 1-1 in Sutton after a thrilling finale with both teams having chances to win it at the death.
This result means that Palace stay in ninth position in the WSL. Next up for Palace Women are Aston Villa on the 17th November at Villa Park.
Palace: Yañez (GK), Veje, Everett, Woodham, Green, Potter, Gejl, Cato, Weerden (Hopcroft, 66), Riley (Larkin, 60) , Stengel
Subs not used: Majasaari (GK), Arthur, Gibbons, Sharpe, Aspin, Atkinson.
Everton: Brosnan (GK), Hayashi, Wheeler, Vanhaevermaet, T. Payne, H. Payne, Finnigan, Sarri, Holmgaard, Snoeijs (Bissell, 61), Olesen (Lawley, 61)
Subs not used: Ramsey (GK), Settle, Foster, Hobson, Thomas.