The South Australian, who recorded the biggest win of his career when he beat Roger Federer in Miami last month, was handed a wild card to the tournament, but lost 7-5 6-4 to Russian world No.40 Karen Khachanov.
Watch - Kokkinakis get his revenge after taking a fall:
Mischa Zverev overcame a mid-match blip to defeat Canadian teenager Felix Auger-Aliassime 6-2 6-7(4) 6-1.
German Zverev converted six out of 14 break points against the world No.176 to prevail in two hours 34 minutes and set up a meeting with French seventh seed Lucas Pouille.
Former world No.1 Novak Djokovic, who is struggling to return to his best form since having elbow surgery earlier this year, takes on Serbian compatriot Dusan Lajovic on Monday.
The 12-times grand slam winner ended his collaboration with head coach Andre Agassi and coach Radek Stepanek earlier this month.
World number one Rafa Nadal, who returned to action from a leg injury in Spain's Davis Cup win against Germany last week, is chasing a record-extending 11th title at the claycourt tournament.
