Compton's highest score for England in his last eight innings was 57 during the recent Test series victory in India, and there have been doubts cast over his place in the side.
However, Cook has given the Somerset batsman a vote of confidence ahead of Wednesday's first Test in Dunedin.
"I'd just love to see him get in and get that one big score which will get him up and running," Cook told BBC Sport.
"He's definitely got the class to do that at international level. One of his great strengths, when he plays for Somerset, and the reason he forced his way into this England side, was that when he got in he went big.
"It was slightly unusual that didn't happen in India, but I've got no concerns about that, because that's why he got picked. He's got the ability to get in and score big hundreds."
Compton has a 34.66 average in Tests for England.