Drago, 46, beat Welshman Jones 5-1 to set up a meeting with Alan McManus in Thursday's third round at the World Snooker Academy.
Joining the Maltese in the next round will be fellow old-timer James Wattana, who beat Fraser Patrick 5-2.
Kurt Maflin, Rod Lawler and Barry Pinches were among the day's other winners in Yorkshire.
Liam Highfield was also a victor - seeing off Ian Burns 5-2 - and moves directly into the fourth and final round thanks to a walkover against his next scheduled opponent Jimmy White.
Competition comes to a head on Friday, with the 16 winners booking places at the tournament proper in Bendigo, Victoria next month.
Round two results:
Adam Duffy 3-5 Thepchaiya Un-Nooh
Passakorn Suwannawat 2-5 Mitchell Mann
Kurt Maflin 5-4 Robbie Williams
Craig Steadman 1-5 Thanawat Tirapongpaiboon
Simon Bedford 2-5 Michael Wild
Alfie Burden 2-5 Cao Yupeng
Andy Hicks 3-5 Sam Baird
Yu De Lu 2-5 Michael Wasley
Dechawat Poomjaeng 5-4 Tian Pengfei
James Wattana 5-2 Fraser Patrick
Liam Highfield 5-2 Ian Burns
Tony Drago 5-1 Duane Jones
Barry Pinches 5-4 Aditya Mehta
Li Yan 1-5 Pankaj Advani
Mark Joyce 0-5 Rod Lawler
David Grace 3-5 Martin O'Donnell