Allen made breaks of 51, 53 and 122 in his comfortable victory over the English amateur on Tuesday.
Five-time world champion O'Sullivan clinched a 4-1 win over his English compatriot David Gilbert.
Current world champion Mark Selby, Judd Trump, Liang Wenbo and Alfie Burden also progressed to the last 16.
Mark Allen set up a last-16 meeting with Ronnie O'Sullivan at the European Masters in Bucharest on Wednesday as he earned a 4-0 win over Daniel Womersley.