Runs of 69, 79 and 70 saw Robertson into a 4-0 lead and it looked as though it would be a five-frame advantage when he made 59 at the start of the next but Carter managed to grind out the frame 64-59 to cut the arrears.
Carter then constructed a 106 to further close the gap but Robertson took frame seven with a break of 67 before seeing out the contest with a 41 clearance.
Robertson, who had struggled with a virus all week, said: "I had a very good night's rest last night, 12 or 13 hours sleep, which was important and I felt much better today.
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"My performances have improved as the week has gone on and today was the best I have played. It has been a new experience for me to be sick during a tournament but I have kept fighting. It's an amazing achievement."
In Saturday's late session, Ding Junhui whitewashed Mike Dunn 6-0.
"I have kept my momentum going through the week," said Ding. "I have played very carefully during every match.
"I think Robertson has been playing well. I have been watching his matches this week and I don't think he has played badly. He's a much better a match player than most of the others."
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