Humphries – the winner of last year's event – is in good shape to defend his title, but endured a nervy encounter against Wales' Clayton.
Both players split the first two sets before the world number one seized control, helped by a superb 146 checkout when two legs down in set three.
The finish shifted momentum into Humphries' favour as he went on to book his place in the final four.
"That's the most nervous I have felt in a long, long time," Humphries told Sky Sports.
"It is hard playing your friends but I just felt nervous, It is really not like me. There is an opportunity to win again and maybe I am trying too hard. Jonny was unlucky there, he could probably have had the third set, that 146 changed everything.
"I am trying to strive for more and achieve perfection but I am very lucky I am still here. Sometimes you have to ride your luck."
Humphries will come up against Ryan Joyce in the semi-finals after he continued his fine run with a 3-2 win over world number four Rob Cross.
The 39-year-old withstood the pressure of a final-set decider to reach his first World Grand Prix semi-final.
Elsewhere, Belgium's Mike De Decker saw off James Wade 3-0 in a tight encounter that saw each set taken to deciding legs.
De Decker will play compatriot Dimitri Van den Bergh on Saturday after he beat Joe Cullen 3-1 to end the night's quarter-finals.
