Kokkinakis staved off a spirited challenge from Spanish veteran Fernando Verdasco, posting a 6-3 6-7(5) 6-2 win in a centre-court showdown.
The seventh-seeded Australian, who fired 15 aces, bounced back from the disappointment of being unable to serve out victory at 5-4 in the second set. Kokkinakis took control in the deciding group to triumph after two hours and 50 minutes on court.
Next up - Steve Johnson.
Kyrgios rolled past Marcos Giron 6-3, 6-2 in just under an hour. The Aussie was a different class once he settled into the evening session match, saving an early break point before dominating from 3-3 in the opening set.
“I came out a bit slow, obviously very different conditions to Wimbledon,” said Kyrgios, who later admitted it took him a long time to get over his four-set final defeat to Novak Djokovic. “I think it was just a bit of nerves as well. I won the tournament in 2019. Seeing my name up on the stadium and then knowing I didn’t play well here at all last year — I lost first round and I was just going through such a dark time in my life.
“I’m just so happy to be here with my team, the love and support from the crowd. I’m just really happy to be out here playing some good tennis again.”
The Australian will take on Tommy Paul on Wednesday for a place in the quarterfinals. Kyrgios is also playing doubles at the Citi Open this week, partnering Jack Sock.