Murray struggled for almost two and a quarter hours before finally seeing off the world number 91 from Kazakhstan 5-7, 6-3, 6-2 in an enthralling match in the Pat Rafter Arena.
Murray arrived on court with his right knee bandaged and appeared to hobble throughout, calling for the trainer twice in the opening two sets and at times seemed to grimace in pain.
An inspired Kukushkin took full advantage and had the top seed in big trouble early as he sent down a barrage of brilliant winners from all over the court.
Kukushkin broke Murray twice and raced to 4-0 and 5-1 leads before Murray at last began to find his range, fighting back to 5-5 only to lose his serve and hand the first set to his opponent.
However, Murray slowly began to take grip of the match, breaking Kukushkin early and holding on to level the contest at one set apiece.
In the decider Kukushkin finally ran out of steam and a relieved Murray closed out the match to the delight of a packed centre court.
"He was hitting winners from all over the place," Murray said.
"I was a little bit slow but I wasn't expecting him to come out swinging like that.
"He was hitting the ball great so I was glad I was able to turn it around.
"You're meant to be ready from the start and I wasn't quite there today," he added.
"But I played better towards the end, I started to move a little bit better. My body loosened up a little bit so hopefully I will pull up okay tomorrow."
Japan's Kei Nishikori rallied from behind in his match, losing the first set and being 4-1 down in the third to overhaul German Cedric-Marcel Stebe 3-6, 6-1, 6-4.
The fifth seeded Nishikori will now take on crowd favourite Marcos Baghdatis of Cyprus.
The 22 year-old said he had watched Baghdatis play on Monday night and was expecting a solid performance from the man he acknowledged as "always a good player".
Nishikori was joined in the second round by fellow countryman Tatsuma Ito, who accounted for Australian wildcard Ben Mitchell 6-4, 6-2.
Former champion Radek Stepanek also played and beat an Australian wildcard, Matthew Ebden, although the sixth seed had to fight all the way before winning 6-4, 3-6, 6-3.
Seventh seed Jurgen Melzer of Austria was beaten 6-1, 7-5 by German Philipp Petzschner, while other winners included Fin Jarkko Nieminen, Colombia's Santiago Giraldo and Russian qualifier Igor Andreev, a 7-6 (8/6), 6-2 winner over Frenchman Adrian Mannarino.