The Robins looked to be heading for just their second defeat in Hellenic Division One East after Billy Gordon gave Woodley the lead late in the second half.
But Williams levelled the match in the 96th minute when he cooly converted from close range.
Elsewhere, Wokingham & Emmbrook were dumped out of the Hellenic Cup after suffering a surprise 2-1 defeat at Premier Division basement boys Abingdon Town.
Tom Larman scored both goals for Abingdon either side of a strike from Elliott Rushforth after 65 minutes.
Carl Davies scored a brace in Binfield's 3-1 win over Tytherington Rocks in the Hellenic Cup.
The striker is set to undergo a hernia operation soon and will be out of action until around Christmas.
If Saturday's performance is anything to go by, the Moles will sorely miss the influential striker, who as well as scoring twice against Rocks, set up Joel Havermans to head home.
Finchampstead beat Hellenic Division One East strugglers Henley Town 2-0 thanks to goals from Harry Swabey and Stewart Manicom.
Sandhurst remain second from bottom in Combined Counties Premier Division after losing 3-2 at home to Croydon.
Meanwhile, Eversley & California slipped down to fifth in Division One following a 3-0 defeat by high-flyers Frimley Green.
* SEE THURSDAY'S NEWS FOR FULL REPORTS AND PICTURES.