The amateur, once ranked 10 in the world and a twice semi-finalist at the Crucible, fell 10-7 to end his run of six straight victories at the English Institute of Sport.
Joyce, the world number 46, moves on to meet Michael Holt for a place at the main event, which starts on April 20.
Alan McManus, another to have twice reached the final four at the Crucible, moved to within one win of securing his place at the tournament again with a 10-8 victory over Nigel Bond.
But there was heartbreak for veteran Tony Drago, the Maltese falling 10-8 to David Gilbert.
The final round of qualifying takes place over the weekend, with the ever-popular Jimmy White and former world champions Ken Doherty and Peter Ebdon among those in action.
Round seven results
Mark Joyce 10 - 7 Joe Swail
Rod Lawler 10 - 9 Anthony McGill
Sam Baird 10 - 9 Rory McLeod
Yu Delu 10 - 6 Jamie Burnett
Liam Highfield 10 - 9 Jamie Jones
Dechawat Poomjaeng 10 - 4 Anthony Hamilton
Alan McManus 10 - 8 Nigel Bond
David Gilbert 10 - 8 Tony Drago
Liang Wenbo 10 - 3 Jimmy Robertson
Alfie Burden 10-9 Dave Harold
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- Tony Drago