Williams and John Higgins will meet in a mouthwatering quarter-final tie at the World Championship, while fellow Class of 92 graduate Ronnie O'Sullivan is also on course for the last eight.
Williams resumed his last-16 clash with Vafaei nursing a two-frame advantage, and he continued to keep the Iranian at arms lengths until finishing the match with a sublime 115.
In a contest which started poorly but improved through the three sessions, Williams always appeared to have matters in control, even when Vafaei put together breaks of 74 and 72 to pull the score back to 11-10.
Williams, bidding to be world champion for a fourth time, remained typically unflustered and put himself within touching distance of victory by dominating frame 22, before finishing the job in style to complete a 13-10 triumph.