"Alastair was a gentleman and a great ambassador for the game of cricket," Clarke told News Corp Australia.
"I've always believed it's not what you say, it's what you do and he's a great example of that.
"He wasn't a big talker on or off the field but when he said something it was important, and that was the way he played."
Cook led England to two home Ashes series wins over Clarke's Australia but also suffered a 5-0 defeat Down Under in 2013-14.
The 33-year-old left-hander was often criticised for his captaincy style, and his batting form, a level of scrutiny that Clarke said was unwarranted.
"He was a fighter and he batted in what I believe is the toughest position in our game, opening the batting," Clarke added.
"He found a way to score runs in all conditions and that's a sign of a great player, that he can adapt and find a way to be successful all around the world. He probably hasn't received the credit he has deserved for being such a good player."
Cook's finest hour came during the 2010-11 Ashes series when he scored 766 runs to help England win their first series in Australia for 24 years.
Former Australia fast bowler Peter Siddle said Cook's ability to adapt to different conditions was admired even by the opposition dressing-room.
"It could be incredibly frustrating trying to bowl at him when he was playing at his best — and he was amazing during that series," Siddle told the Times.
"He would take the same approach if he was under pressure or if he had been dominating our bowlers. Even from the opposition dressing room, that was something you had to admire."
(Reporting by Hardik Vyas in Bengaluru, editing by Ed Osmond)
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