The former pro, once ranked 10 in the world, registered his sixth consecutive victory at the English Institute of Sport, 10-6 over the world number 59.
Swail, who has reached the World Championship semis twice - in 2000 and 2001 - and the last 16 a further four times, is now just two wins away from securing a spot in the main draw at the Crucible.
He is the only amateur remaining in contention to qualify for the tournament, which starts on April 20, and next faces Mark Joyce, ranked 46.
Swail was one of three former Crucible semi-finalists in action on Tuesday.
Alan McManus, who reached the last four in 1992 and 1993, advanced after a 10-9 win over Aditya Mehta but Andy Hicks, semi-finalist in 1995, dropped to a 10-7 defeat to veteran Tony Drago.
McManus now plays Nigel Bond while Drago meets David Gilbert.
Round seven also sees the likes of 55-year-old Steve Davis, who has won the tournament six times, enter the fray.
Round six results:
Rod Lawler 10-5 Fraser Patrick
Tony Drago 10-7 Andy Hicks
Jimmy Robertson 10-2 Li Yan
Alan McManus 10-9 Aditya Mehta
Dechawat Poomjaeng 10-9 Liu Chuang
Joe Swail 10-6 Adam Duffy
Yu Delu 10-2 Ian Burns
Liam Highfield 10-9 Barry Pinches
Sam Baird 10-9 Peter Lines
Alfie Burden 10-7 Paul S Davison