Root led England to a 211-run win over South Africa at Lord's in his first Test match as captain since succeeding Cook, who resigned in February after a record 59 Tests in charge.
Cook was impressed by the way Yorkshire's Root, 26, handled himself.
"By playing it a bit more low-key, certainly in the dressing room, his message will grow," said Cook.
"It will get clearer and stronger as he goes."
Listen: Joe Root - England captain
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Root, who is known for his dressing room spirit, had only captained in four previous first-class matches before taking on the England job.
"You often see a captain who comes in and thinks he's got to do the speech at the start of his reign, saying that this is what he wants and stuff," added Cook.
"Then you've forgotten everything he's said after two minutes.
"But actually because he didn't do that, everyone was still waiting for it and when he did speak, he spoke very clearly and very concisely on what he wanted to do."
Cook, 32, does not think that Root's personality will be dampened by the new role.
"I still think you'll see the glint in his eye," he said.
Some questioned whether Cook would struggle under a new captain at Test level.
However, the Essex batsman thinks his relationship with Root will continue to flourish.
"As he feels more comfortable in the role and I feel more comfortable, I will maybe throw a few more suggestions at him, but in the first Test it was probably business as usual," added Cook.
"We were both still at first and second slip. It was me bouncing ideas off Joe and him deciding which idea he might take or not.
"You go back a year and it was him giving me the ideas and me making the decisions."
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Cook said he is not contemplating retirement from the Test stage and wants to carry on playing for as long as possible.
"I genuinely love playing cricket, " he said. "I genuinely love, whether it's an Essex shirt or an England shirt, trying to score runs and trying to set up wins for the side.
"Hopefully I can be part of it for as long as I can, because one day some bloke will tap me on the shoulder and say 'we don't want you any more'."
Alastair Cook has backed the "low-key" approach adopted by Joe Root, his successor as England Test captain.