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At 63-63, Robertson took the first frame with a respotted black but Carter took the second with a 94 break.
Carter won the third frame to go 2-1 up but Robertson fought back with a break of 43 and 57 to take a 3-2 lead.
However, Carter came from behind to seize victory, as he won three straight frames including an 84 break.
He will face off against Trump, who beat Ronnie O'Sullivan on Wednesday, in Friday's semi-final.
Semi-finals
Judd Trump v Ali Carter
Mark Selby/Marco Fu v Ding Junhui/Anthony Hamilton
Quarter-finals
Mark Selby v Marco Fu
Ding Junhui v Anthony Hamilton
Neil Robertson 3-5 Ali Carter
Ronnie O'Sullivan 3-5 Judd Trump
Round one
Stuart Bingham 1-5 Anthony Hamilton
Ronnie O'Sullivan 5-1 Liang Wenbo
Mark Selby 5-4 Ryan Day
Marco Fu 5-1 Anthony McGill
Barry Hawkins 0-5 Neil Robertson
Judd Trump 5-2 Mark King
Ding Junhui 5-4 John Higgins
Ali Carter 5-4 Shaun Murphy
Ali Carter beat Neil Robertson 5-3 in the Players Championship quarter-final in Llandudno, to set up a semi-final tie against Judd Trump on Friday.