Williams – who won the event in 1996 and 1999 – will face 2018 runner-up Barry Hawkins in the third round after the Englishman produced breaks of 69, 72 and 52 in a 4-2 triumph over Pang Junxu.
Robertson – Welsh Open champion in 2007 and 2019 – came from a frame behind to beat Vafaei.
The Iranian came out on top in an opening frame that lasted 51 minutes, although former world champion Robertson won four consecutive frames, the second of which was on the black, to claim victory.
Having survived those early scares, Robertson showed his class in the final two frames with breaks of 100 and 118 to set-up a third-round meeting with Jak Jones.
The Welshman hit a trio of 50-plus breaks to move into round three with a 4-2 win over Scotland’s Anthony McGill.
A stunning 101 break in the penultimate frame proved pivotal as Welshman Jackson Page came from behind to beat Chang Bingyu 4-3.
Bingyu posted breaks of 65 and 74 en route to going 3-2 up, although Page’s century break was followed by a 70-22 win in the final frame.
Robbie Williams came from 2-0 down to beat 2013 winner Stephen Maguire 4-3 in the second round – with a 69 break in the fourth frame the highlight for the Englishman.
Jack Lisowski won three successive frames to come from 3-1 down to beat Xiao Guodong on the north Wales coast.
China’s Guodong looked on course for victory after following up a pair of half-century breaks with a 113 in the fourth frame.
But Lisowski produced an impressive fightback to win each of the final three frames, producing an 83 break to force a last-frame decider, which he then went on to win 56-39.
David Grace claimed a straightforward 4-1 victory over Welshman Dylan Emery, while Wu Yize defeated Joe O’Connor 4-2 in a match that featured four half-century breaks.
Elsewhere, Jiang Jun outscored fellow Chinese player Yuan Sijun to secure an emphatic 4-0 victory.
