Kokkinakis ends Kudla's 9-match win streak and equals his career-best 4th round appearance at a Masters 1000 event.
Kokkinakis has to dig deep to reach the final 16 of an ATP 1000 Masters event for the first time since 2015.
The 25-year-old converted just one of seven break points but managed to edge both tie-breaks with his 21 aces proving decisive.
Kokkinakis will now face world number four Alexander Zverev in the next round on Wednesday with the German sweeping past Mackenzie McDonald 6-2 6-2 in their round of 32 clash.
It's been a resurgent year for Kokkinakis, who won his first men's singles ATP title in January at the Adelaide International before winning the Australian Open men's doubles title alongside Nick Kyrgios.