Carter showed his intent with a 130 total clearance in the first frame and doubled his advantage with a 57 break.
Essex's Carter won a scrappy third frame 53-21 and made it 4-0 with a 67 break.
After the interval, former world champion Ebdon missed a tough opening red to the middle pocked from tight on the baulk cushion. Carter then stepped up with his second century of the match to leave Ebdon needing to win six successive frames.
It was a task that was beyond Ebdon and Carter sealed a deserved win with a 64 break.
Earlier, Hendry admitted he did not deserve to beat Robertson, who will play Scotland's Stephen Maguire in the last eight.
"When you make loads of unforced errors you don't deserve to win," said the former world champion.
"When you feel as uncomfortable as I do out there results aren't going to happen."
It was 2-2 at the mid-session interval between Robertson and Hendry with two scrappy opening games being followed by an 85 break for Robertson and then an 80 for Hendry in the fourth frame.
Robertson will play Stephen Maguire in the last eight
Robertson had a 147 in his sights after the break but missed an awkward black on 97.
The Australian should have been 4-2 ahead when he opened the sixth frame with a 52 break but Hendry hit back with a 61 to steal the frame on the black.
Hendry edged ahead in the next frame, getting the better of Robertson in a protracted safety duel. However, it was all square again after the eighth frame.
Robertson scraped home with a 47 break in the next and the left-hander ended the seven-time world champion's participation at Wembley with a 92 break in the last frame.
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