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With the scores at 8-8 heading into the afternoon session, the 2010 champion regained the lead with a break of 59 before he took the 18th frame as Hawkins missed a costly red.
'I need to be patient' - McGill on defeat to Bingham at Crucible
Robertson took the 19th with a half century before Hawkins pulled back one with his only century of the afternoon.
In the end, though, it proved to be a relatively comfortable conclusion to the contest for Robertson.
The Australian responded with a 79 clearance after a lengthy 21st frame before sealing his place in his eighth World Championships quarter-finals where he will face Mark Selby.
'Just can't wait' - Emotional Bingham on reaching semi-finals
'Brilliant pot' - Trump sinks outrageous long red from cushion
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