The big-serving Russian had five aces and did not face a single break point.
Kokkinakis recently beat 20-time Grand Slam champion Roger Federer in the second round of the Miami Open -- becoming the lowest-ranked man to beat a No. 1 in 15 years -- but the powerful Australian could not produce anything like that form this time.
Khachanov constantly pressured his opponent's serve, yet could convert only two of his 13 break points.
If he beats the winner of that all-French match, Khachanov could face defending champion Rafael Nadal in the third round.
Zverev labored to a 6-2, 6-7 (4), 6-1 win against 17-year-old Canadian Felix Auger-Aliassime and next faces seventh-seeded Frenchman Lucas Pouille.
Medvedev rallied to beat Marton Fucsovics of Hungary 6-7 (4), 6-2, 7-5, saving seven of eight break points. He meets either Kei Nishikori of Japan or 12th-seeded Czech Tomas Berdych, the 2015 runner-up, on Monday.
Two-time champion Novak Djokovic, still struggling with a persistent right elbow injury, is also in first-round action Monday.
The ninth-seeded Serb, who recently split with coach Andre Agassi, takes on countryman Dusan Lajovic. He beat Lajovic in their only previous meeting three years ago.