Trump took steps to even out the match to make it 5-5 yet Carter edged a strained 10th casing to seal triumph.
World number eight Carter will meet Northern Ireland’s Imprint Allen or Welsh Open champ Gary Wilson in the semi-finals.
Prior Imprint Selby beat Barry Hawkins 6-3 to set up a quarter-last tie against Ronnie O’Sullivan.
Selby brought off four edges in succession to move beyond Hawkins, who had taken a 2-0 lead.
“Barry let me free a little. He had a decent opportunity to go 3-0 and he missed a red you wouldn’t anticipate that he should miss, and the game kind of turned on that. I played some great stuff after that,” Selby told BBC Radio Shropshire.
“Each time I play Ronnie is consistently an extraordinary event. I lost to Ronnie in the semis here a long time back. Neither of us are getting any more youthful. More silver hairs, so I guess we need to congratulate ourselves for as yet being here.”
China’s Zhang Anda beat Thailand’s Noppon Saengkham 6-5 to finish the quarter-last line-up, where he will meet John Higgins.
Zhang came from 5-4 behind and fixed the success with a 65 break in the concluding casing of a match which endured more than four and a half hours.