Draper needed only one hour 20 minutes to complete a 6-1 6-4 victory over American Paul in the quarter-finals in Adelaide.
Having powered into a 4-0 lead, Draper broke for a third time at 5-1 to take the opening set.
Paul, who also lost to Draper in Adelaide last year, came out stronger in the second set and managed to hold serve until Draper got the decisive break at 5-4 and secured the win in style with a superb passing shot down the line.
Draper will play Kazakhstan’s Alexander Bublik in the semi-finals, while Sebastian Korda of the US faces Czech seventh seed Jiri Lehecka.
In the women’s draw, Kazakhstan’s world number three Elena Rybakina lost 6-3 6-3 to Russia’s Ekaterina Alexandrova, who will face Jelena Ostapenko in the semi-finals after the Latvian sixth seed beat Ukraine’s Marta Kostyuk 7-5 6-3.
American second seed Jessica Pegula beat Russian Anastasia Pavlyuchenkova 6-7 (1-7) 7-5 6-4 to reach the semi-finals.
Pegula will play Russian world number 15 Daria Kasatkina, who progressed after Laura Siegemund withdrew.
Elsewhere, American top seed Ben Shelton reached the semi-finals of the Auckland Classic by beating Spain’s Roberto Carballes Baena 6-4 6-3.
At the Hobart International, Belgian top seed Elise Mertens defeated the Netherlands’ Arantxa Rus 7-5 6-0 to set up a last-four meeting with Australia’s Daria Saville.
British number three Andy Murray won his first match of the 2024 season by beating US Open champion Dominic Thiem 6-4 6-2 in the Kooyong Classic exhibition event.
Three-time Grand Slam winner Murray was knocked out in the first round of the Brisbane International this month and lost to Marin Cilic at Kooyong on Wednesday.
Murray has been drawn against 30th-seeded Tomas Martin Etcheverry of Argentina in the first round of the Australian Open.