Hendry and Davis book semi spots.
But, with his back against the wall in the 16th frame, the five-time winner of the event produced a polished break of 64 to close out the match 9-7.
Hendry led the world number 19 5-3 at the halfway point of their clash, with the talking point of the first session being Carter's spurned 147 attempt. The Essex man rattled the jaws of the pocket with the 14th black, ruling himself out of the running for the pounds 26,000 prize for a maximum.
Hendry was nowhere near as fluent as he was in the third-round match against Mark Williams, with the 76 he made in frame one his highest score.
Steve Davis - the man whose record six world championship titles was eclipsed by Hendry - extended his fairytale tournament when he prevailed 9-7 over Ireland's Ken Doherty.
Davis was also coasting at 8-4 but was dragged back to 8-7 by the redoubtable Dubliner. Having got in first for the third straight frame, however, Davis finally made it pay in a tense clincher.