Steve Jones supplied the cross which was met by Higdon's firm header to cancel out Clinton Morrison's near-post header which had put Palace 2-1 up.
It was Crewe who had opened the scoring, Jones prodding the ball home after only six minutes.
Darren Ward hit the post as Palace responded before Dougie Freedman stole in at the far post to score the match's first equaliser on 52 minutes.
Crystal Palace boss Iain Dowie:
"It's frustrating but you've got to give Dario Gradi credit.
"In the first 10 minutes and the last 15 they did great but outside that we should have wrapped the game up.
"We had chance upon chance and the goal in the first half was their only shot on target - but they're a good football side,"
Crewe: Williams, Bignot, Foster, Billy Jones, Tonkin, Gary Roberts, Cochrane (Rivers 90), Lunt, Vaughan (Rodgers 80), Steve Jones, Higdon.
Subs Not Used: Tomlinson, Walker, Johnson.
Booked: Cochrane.
Goals: Steve Jones 7, Higdon 90.
Crystal Palace: Kiraly (Speroni 59), Butterfield (Borrowdale 45), Ward, Fitz Hall, Boyce, Reich (McAnuff 86), Watson, Soares, Hughes, Freedman, Morrison.
Subs Not Used: Hudson, Andrews.
Booked: Fitz Hall.
Goals: Freedman 52, Morrison 65.
Att: 6,766
Ref: M Dean (Cheshire).