The Robins have won six of their 10 League One games since former Brighton Under-21 and reserves coach Williams took interim charge in December.
The 35-year-old replaced Martin Ling at the County Ground, after Ling stepped down citing "health reasons."
"He is a fantastic coach, the best I've come across in 25 years of football," chairman Lee Power said.
"The reason Luke is 'head coach' and not 'manager' is because that is what he is: the head coach.
"No players will come into the club that Luke is not happy with, but his main job is to coach the group and improve them."
Williams is the fourth person to take charge of Swindon this season, following Mark Cooper, Ling, and chairman Power, who looked after the first team for a spell following Cooper's dismissal in October.
The Robins are currently 13th in the League One table, seven points outside the play-off places.
Swindon Town have appointed caretaker boss Luke Williams as head coach on a five-year contract.