Norrie claimed his biggest career title at the hard-court tournament in 2021 but was beaten this time by home hope Paul, who will play Russian fifth seed Daniil Medvedev next.
“It was a super weird match, a lot of breaks,” said Paul, the world number 11.
“I started pretty well and tried to play some aggressive tennis. It was an up-and-down match on both sides.”
In their first meeting since 2022, both players struggled on serve in the first set.
Norrie, ranked 77th in the world, was broken three times before handing Paul the opener with a double fault on the second set point.
In the second set, Norrie again found himself a break down at 4-2 but rallied to level before taking a 0-30 lead on Paul’s serve.
Paul fought back before taking advantage of Norrie’s error-strewn serving, the Briton conceding the match with his fifth double fault of the set and seventh of the match.
Draper is left to carry British hopes in the men’s draw, with the 13th seed set to face American Jenson Brooksby on Monday.
Medvedev progressed after his unwell opponent, American Alex Michelsen, retired just two games into their match, while Greece’s Stefanos Tsitsipas beat Matteo Berrettini of Italy 6-3 6-3.
