Murray began in impressive fashion and his blistering start included two double-breaks to take the first set 6-1. Kukushkin could not handle the Scot's aggressive style that dominated the court.
The second set started where the first left off, with Murray breaking serve again to race into an early lead. Kukushkin struggled to deal with Murray's powerful cross-court forehand time and time again.
Kukushkin did eventually break serve, but a revival was short-lived as Murray quickly took two sets without reply. The footwork of the 24-year-old Scot continued to cause all kinds of problems as Kukushkin looked increasingly drained. Murray served for the set to take it 6-1.
The third set began with Kukushkin moving uncomfortably. Murray went 1-0 up before Kukushkin was forced to retire through injury.
Murray will now face either Jo-Wilfried Tsonga or Kei Nishikori in the next round.