The world number 16 defeated 2003 champion Matthew Stevens at the Barbican in York, continuing his fine 2012 as he reached his third ranking event semi-final of the season.
Davis, a thrilling 6-5 victor over world number three Higgins in the previous round, will now face Mark Selby in Saturday's second last-four clash.
After beating Stevens 6-4, the Sussex potter said even he was surprised at how relaxed he had been, considering the importance of the match.
"I actually felt quite calm," he said. "I knew it was a massive game so I was quite surprised myself to be honest with you.
"It is a brilliant win for me and I can't wait to play the semi-final.
"I don't know what is going on with me playing so well. Getting into the top-16 for the first time at 40 is probably some sort of record I would imagine.
"I'm just playing well. It is the playing opportunities we have got nowadays because we are playing so much.
"I haven't had a chance to think about winning the tournament yet even though I suppose I am quite close, but there is a lot of frames to be won yet."
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