The emotional and physical impact of the past week was always likely to catch up with Draper, but it is a testament to his improved fitness that he kept the match so close.
He and Paul traded breaks at the start of the first set but a poor serve and volley cost Draper, allowing his opponent to break for 5-3 and subsequently serve out the set.
Draper grew into the second set, upping his aggression behind his forehand in particular, and had a set point at 5-4 before Paul responded with three huge serves.
Paul cracked when serving to try and force a tie-break, with a long forehand sending Draper into his 17th deciding set of the year.
Draper kept the momentum going, rattling through his service games, but he began to tire, shouting “energy, energy, energy!” to himself as he saved a break point.
However, a poor service game at 4-3 down ultimately cost him, with a fatigued double fault helping Paul secure the decisive break.
“I was definitely having to pick myself up. I felt quite flat at times,” Draper said.
“But at the end of the day, if I want to be one of the best players in the world, then I have to get used to going deep every week.”
Former British number one Andrew Castle told BBC TV: “Just remember – Jack is 22 and there’s a whole load of improvements still to make.
“His best is yet to come and for me that’s the most exciting thing.”
Paul will face fellow American Sebastian Korda after the world number 23 fought back from a set down to defeat Australia’s Rinky Hijikata.