Yorkshire batsman Root will look to guide England to a win that puts them 2-0 up in the four-match series when the second Test gets underway at Trent Bridge on Friday after his first match as captain ended in a dominant 211-run win over the Proteas at Lord's last week.
That was the 26-year-old Root's first match as Test captain since Cook stepped down from the post in February and saw the new skipper star with the bat as well in making a first-innings 190.
"It was a great week for English cricket," Cook told Sky Sports on Wednesday.
"I am really pleased for him. He had a horrible cold all week, but I thought he was outstanding.
"When he did speak in the dressing room his messages were very clear. The confidence he will have got from last week, it is always important to start well.
"He got the monkey off his back in terms of getting a big score early on in your captaincy career, which stops everyone talking about captaincy affecting your batting.
"I think all in all he will have gone to bed Sunday night a very happy man."
No one knows more about the pressures of leading the team than Cook, who captained England in a record 59 Tests before standing down following a 4-0 series loss in India late last year.
- 'Business as usual' -
"Nothing can prepare you for it," Cook said of being England captain.
"Sometimes the peace and quiet is on the field and when you are batting. It is a big role, but it's great that he had a really good start and I am always there if he needs some support.
"Personalities will be different. His will evolve over a period of time. He is still new to captaincy, in terms of any kind of cricket, so he will get better and better. He could not have done more in that first game."
Meanwhile Cook marked his return to the ranks after five years as captain with a valuable 69 in the second innings at Lord's.
"It was business as usual for me," said the Essex left- hander, England's all-time leading Test run-scorer.
"It was a little bit strange, especially when Joe came out to toss and knowing I will never do that again.
"But then you see all the interviews after and think 'it's alright, I can go and get my pads on!'
"I have done my stint, proud of what I achieved and hopefully we can build forward with Rooty."
Cook added: "The target hasn't really changed for me. It is just to score runs to help set up England victories... Scoring big hundreds is the dream for any batter."
South Africa have lost just one away Test series in the last 10 years and Cook expected the Proteas to bounce back strongly at Trent Bridge.
"You watch a lot of their cricket and they are one of the sides that travel very well," he said. "They have a very good record away from home. They will be hurting.
"We need to be right on it. In Test-match cricket if you are not on it for the full five days you can lose it very quickly.
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