The 27-year-old Greek Australian saw off British 20th seed Evans 6-4, 6-4, 6-4 to win on the Paris clay for the first time since reaching the last 32 as a teenager in 2015.
Kokkinakis will face either former champion Stan Wawrinka or veteran Spanish clay-courter Albert Ramos-Vinolas in round two.
If he was to make a remarkable run to the final, like his good friend Nick Kyrgios did last year at Wimbledon, he may then have to beat 11th seed Karen Khachanov, seventh seed Andrey Rublev, Novak Djokovic and Carlos Alcaraz.
"Someone sent me my path to the final, obviously ambitious, but it's one of the toughest draws I've ever seen," said the world number 108.
"But I probably wasn't expected to win today either so I'll rest up, recover and see what I can do."
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