Quality chances were few and far between in the second-half, as neither team could consistently break down the other.
16-year-old Drakes-Thomas almost opened the scoring for the Eagles on 68 minutes, but his left-footed effort deflected off the defender for a corner.
Substitute Jack Bray could have found the net for the hosts on 79 minutes, but powerful defending from Luke Browne kept it scoreless.
Legs began to tire late in the second-half, and there were chances for both teams to win it in the final minutes.
Two Palace substitutes combined on the 86th minute to almost grab a late winner, as Rylan Brownlie slipped in Sebastian Williams.
Despite a cute first touch from Williams, West Brom goalkeeper Okonkwo was equal to the effort and kept the scores level.
There was late drama as Gibbard was shown a straight red card on 90 minutes, after his reaction to a foul from a West Brom defender.
With seconds remaining on the clock and Palace a man down, West Brom had the best opportunity of the game in the 93rd minute.
Dan Chimeziri was left unmarked from a West Brom corner, but the striker could not get his header on target.
A point at the Hawthorns leaves Palace in 11th place in Premier League 2, with a trip to Bromley in the Football League Cup awaiting Powell's side on Tuesday, 2nd September.
**West Brom:** Okonkwo (GK), Nelson, Humphries, Crowther, Diomande (Mohammed, 46), Dupont, McDonald (Chimeziri, 77), Mandey (Bray, 57), Ranger (Mfuamba, 57), Higgins, Ntege
**Subs not used:** Brady (GK)
**Palace:** Lee (GK), Browne, Reid, Nascimento (Williams, 81), Jemide, Gibbard, Grante, Whyte, Marsh, Abginone, Drakes-Thomas (Brownlie, 81)
**Subs not used:** Izquierdo (GK), Dashi, Walker-Smith