The 2005 world champion defeated close friend Carter 9-8 having trailed 8-4 at the evening mid-session interval.
But Murphy returned to rattle off five straight frames and set up a meeting with either Mark Selby or Mark Davies in Sunday's final at the Barbican in York.
The 30-year-old was at a loss to explain his stunning revival, but says he feels ready to add to the solitary UK Championship title he won in 2008.
"Maths was never my strong point, and I like it when things are simple," admitted the world no.4.
"I was 4-0 down with five to play and that is pretty straight forward, you know where you stand with that - there is no room for error.
"My attitude wasn't any different. I tried to play the right shots and luckily they went in.
"I have been behind further but for the semi-finals of the UK Championship, when I was all but dead and buried, that has got to be my best five-frame spell ever.
"It is a tournament I have done well at in the last few years, and to have another crack at the trophy is very, very exciting."
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