Cook's innings was his fifth century in five matches as captain, while he also became the youngest player, at 27 years and 347 days, to reach 7,000 Test runs in the process, beating India legend Sachin Tendulkar's record.
Compton added: "It says a lot about his hunger and determination, the fact that he's been so consistent over a period of time at a young age.
"A lot of batsmen only really find their feet at 28, 29, and the fact that he's found his feet at a very young age and done it for so long is amazing.
"The stats speak for themselves - to be ahead of Sachin Tendulkar at this stage is an amazing achievement."
Former England captain Michael Vaughan said Cook had impressed him since first walking into the England dressing room and backed the Essex man to surpass the records of Test greats such as Ricky Ponting and Rahul Dravid.
He added: "He's a terrific player. It's quite bizarre to think he's got 7,000 runs and 23 centuries and he's only just starting to get recognition.
"He's certainly going to break all of the English records. He will become a great.
"In Alastair Cook, England have got an outstanding young man and we're in safe hands for many, many years."
Cook's cricketing school days
Derek Randall, former England international and Cook's cricket coach at Bedford School:
"You'd turn up in the morning and he was always first there and last away, and he gave 110% the whole day. I remember the first time I saw him on the bowling machine and I just couldn't believe the balance and the time he had to play."
Guy Fletcher, director of sport:
"He was a frail and small 14-year-old but it was apparent early on that we had a very talented lad on our hands."
Andrew Morris, head of music:
"He's got that sort of choir boy, cherubic look, but underneath that he wants to win and that's a great attribute to have. That's why he is where he is today. He was a natural musician and of course I'd love to have seen him be that but I'm extremely happy that he's captain of England."
Geoffrey Boycott, one of four England batsman with 22 Test centuries to his name - alongside Walter Hammond, Colin Cowdrey and Cook's team-mate Kevin Pietersen - says Cook will not be focusing on the achievement.
"I don't think it will be such a big deal to him," Boycott said. "At his age he's going to get quite a lot more unless he has a serious illness or injury.
"He's in the best years of his life and will automatically play, he has a good technique and should be well on the way to 40 hundreds by the time he's finished.
"The great side to him is the fact that he doesn't allow any bowler to get himself out of his comfort zone. He plays perfectly as an opening batsman.
"He has looked in no trouble whatsoever (in Kolkata); he's got these spinners in his pocket. I haven't seen at any time that this lot look like they will get him out."
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