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Carter, who knocked out champion Stuart Bingham in the first round, led 5-3 and 9-7 after the first two sessions.
McManus won four consecutive frames to lead 11-9 and, although Carter made a century to trail 12-11, the Scot took the one he needed.
Mark Williams beat Michael Holt 13-8 to reach the quarter-finals.
Two-time champion Williams will play Judd Trump or Ding Junhui on his first appearance in the last eight since 2011.
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Five-time champion Ronnie O'Sullivan trails Barry Hawkins 5-3 after the first session of their second-round match.
O'Sullivan made a stunning 139 break in only eight minutes in the second frame but Hawkins won the last two. They resume on Sunday at 14:30 BST.
World number one Mark Selby leads Sam Baird 9-7. They will continue at 19:00 BST on Saturday.
Carter, who suffered several kicks and heavy bounces off the cushions, described the table as "the worst I have ever played on".
"The conditions were horrific. There were kicks every other shot and big bounces off the cushions," said the Englishman.
"But it was a disgusting performance from me. Alan dictated the match and I was powerless to do anything about it.
"I just wanted to smash the place up at the interval."
Read more: Organisers defend 'horrific Crucible tables
McManus has reached the quarter-finals in Sheffield five times in 20 appearances since his debut in 1991, reaching the semi-finals twice.
He failed to qualify between 2007 and 2012 but will face Ricky Walden or John Higgins in the last eight.
"I got the ideal start on Saturday having been 9-7 down," said 45-year-old McManus, the oldest player left in the tournament.
"I won a frame on the black ball and relaxed after that. I wanted to be 10-10 but 11-9 ahead was huge for me.
"I still practise a lot and work hard at the game. I enjoy it. Sometimes I am out there looking around and think 'I might not be here again - it is a special place to be'."
Two-time finalist Ali Carter suffered a surprise 13-11 defeat by Alan McManus in the second round of the World Championship in Sheffield.