The world number 19 saw off his opponent 6-3 6-7 6-1 in just under two and a half hours to set up a quarter-final clash with Chile's Alejandro Tabilo.
Having claimed the opening set, Norrie had match point in the tie-break before Van Assche came through to bring the match level.
He was then broken by Van Assche in the opening game of set three before hitting back emphatically, rattling off six games in succession to secure the win.
Norrie is seeded 19th for the Australian Open ahead of the draw taking place in Melbourne on Thursday.