The Frenchman jumped out to a two-game lead in the first set. However, his opponent broke back and ran off six consecutive games to seal the first set as Monfils appeared to be suffering from a back issue.
Kukushkin came out on top in the second set, breaking his opponent twice en route to a four-game lead. Monfils was able to hold his third service game of the set, before the Kazakhstani responded in kind to take a 5-1 lead.
Monfils fought back to win four games in a row to level the score, but Kukushkin was able to break back and then hold to win his second set.
Both men secured their first four service games in set three before the Kazakhstani broke Monfils again. However, Monfils was again able to break back, hold and then break his opponent again to win his first set of the match.
The Frenchman maintained his momentum and raced ahead to a three-game lead in the fourth. Kukushkin was able to hold his second service game of the set, but Monfils broke him again and held to tie the score at two sets apiece.
The 14th seed secured a two-game lead in the opening stages of the final set. However, Kukushkin broke back to take a 3-2 lead. Both men were able to hold serve twice more, before the Kazakhstani broke Monfils after a double fault to seal the win.