Resuming with a 12-4 lead in the best-of-25 quarter-final on Wednesday afternoon, the Australian was briefly made to wait as Davis claimed the opener with a 59 break.
But the six-time former champion, at 52 the oldest player to make the last eight for 27 years after beating Mark King and defending champion John Higgins, was merely delaying the inevitable.
Robertson duly sealed a 13-5 success in style with a run of 107 and the Melbourne-born left-hander, who came from 11-5 behind to beat Martin Gould 13-12 in the second round, advances to play either Ali Carter or Shaun Murphy in the semi-final.
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