Fifth seed Mark Williams, who won this tournament in 1999 and 2002, claimed a 6-4 victory over David Gilbert in the evening session to join Robertson in the last 16.
The 50-year-old lost a lengthy opening frame, levelled at 1-1, but a fine break of 102 gave Gilbert the advantage again.
But Williams then recorded breaks of 51, 89, 73 and 55 to move 5-2 ahead and go one frame from victory. World number 24 Gilbert responded with a 100 break for his second century and then a 50 to pull it back to 5-4.
However, it did not matter as Williams did enough to get the victory to set up a match against either China’s 12th seed Xiao Guodong or compatriot Pang Junxu.
Robertson got back into the top 16 in September 2024, ensuring automatic qualification for the major tournaments, by beating China’s Wu Yize 9-7 in the final of the English Open.
The pair will face each other in the last 16 in York later this week after Wu held off a fightback from England’s Michael Holt to secure a 6-4 victory.
Wu, 22, moved into a 5-0 lead, thanks to breaks of 55, 78, 82 and 81, which included 11 reds and 10 blacks before any hope of a 147 ended with a miss on the 11th black.
But 47-year-old Holt, a quarter-finalist last year, staged a recovery with runs of 59, 82 and 60 to pull within one frame, before Wu sealed the success in the 10th frame.
Twelve players from China were among the 32 in York and Wu became the second of those to reach the last 16 after Si Jiahui’s 6-0 triumph over Ryan Day on Saturday.