Draper bounced back from 1-4 down to No8 seed Tommy Paul at the Adelaide International 2 to win five straight games in a 6-4, 6-4 victory. It included four breaks by the British No3, 10 aces and 88 per cent of points won on his first serve.
He now faces Karen Khachanov for the second successive week, the Russian having beaten Draper in their earlier encounter in straight sets, also in Adelaide.
Norrie, meanwhile, playing in his former hometown of Auckland, faced a tougher encounter at the ASB Classic as he defeated Jiri Lehecka 6-4, 6-7, 6-3 in which the top-ranked Brit converted just three of 11 break points on offer.
Andy Murray, meanwhile, picked up his first win of 2023 with victory over Zhang Zhizen in the Kooyong Classic exhibition event on Wednesday.
The former world No1 had been knocked out of his only Australian Open warm-up tournament by Seb Korda at the Adelaide International.
But he recovered from losing the first set to edge out a super tiebreaker against his Chinese opponent for a confidence-boosting 2-6, 6-3, 10-2 ahead of the first Grand Slam of the year.
“He served well and was taking time away but I managed to gradually improve as the match went on,” said Murray afterward.
There were wins for the British women contingent in Australian Open qualifying with Yuriko Miyazaki and Jodie Burrage now just one win away apiece from making the 128-strong line-up for the first Grand Slam of the season.
Miyazaki produced an excellent serving display in a 6-4, 7-6 victory over Korea’s Jang Su-jeong while Burrage was even more clinical in beating Barboca Palicova 6-1, 6-2.
Katie Boulter, however, failed to join her countrywomen in the last round of qualifying after a 6-1, 6-3 loss to Polina Kudermetova.