The two sides had drawn 0-0 on Sunday to force a replay and the game looked to be heading to extra time until Spencer popped up late on to send the home fans into raptures.
Corinthian-Casuals 1-0 Grays Athletic
FA Trophy
Third round qualifying replay
King George's Arena, Surbiton
Wednesday 2 December 2015
Winning sides receive £4,000
Cherno Mendy spooned a first-half Casuals chance over the bar but Josh Uzun had to hook off the line for the hosts after the break as the Ryman Premier Division side looked dangerous.
However, with 94 minutes on the clock, Spencer nodded home Dave Hodges’ cross at the back post to set up a first-round tie with Hungerford Town as two-time winners Grays crashed out.