With Draper's injury reducing him to a groaning wreck, Nadal finished strongly to close out the match 7-5 2-6 6-4 6-1.
With Kyrgios gone and last year's women's champion Ash Barty retired, the chances of a another home winner for local fans are greatly diminished.
Seventh seed Coco Gauff also raced into the second round, while last year's finalist Danielle Collins advanced with a tougher, three-set win over Anna Kalinskaya.
Briton Raducanu will next face teenager Gauff, who won in Auckland and confirmed her good form with a 6-1 6-4 win over Katerina Siniakova.
Italian 15th seed Jannik Sinner, a quarter-finalist last year, was the first man to reach the second round, rolling over Briton Kyle Edmund 6-4 6-0 6-2 on John Cain Arena.