British number one Draper secured the biggest win of his career when he ended the Queen's Club reign of Wimbledon champion Carlos Alcaraz on Thursday.
But he was outgunned by American fifth seed Paul in the last-eight tie in London, going down 6-3, 5-7, 6-4 in a match lasting a shade over two hours.
Draper battled back from a set down but he was broken to love midway through the decider as the match swung Paul's way.
Defeat ended Draper's seven-match winning streak after he lifted his first ATP title in Stuttgart last week.
"I played good tennis and I like the grass," said Paul. "I knew today was going to be a battle.
"It is never easy against Jack. He has been playing such good tennis, so I am really happy to get through that one."
Italy's Lorenzo Musetti continued his impressive grass-court season by ending the run of British wild card Billy Harris 6-3, 7-5 to reach his second consecutive semi-final on the surface.
Musetti, who advanced to the last four in Stuttgart last week, fired 20 winners and committed just six unforced errors.
He is chasing his third tour-level title and will next play Jordan Thompson after the Australian defeated fourth-seed American Taylor Fritz 6-4, 6-3. Thompson, who did not face a break point against Fritz, arrived in London on a five-match losing streak.
In the final match on Friday, America's Sebastian Korda was playing Australian qualifier Rinky Hijikata for the right to face Paul.
