The Yorkshireman took charge of his country for the first time since replacing Cook as captain in February, overseeing a 211-run victory over South Africa in the series opener.
Root led from the front with the bat, hitting a first-innings 190 to set up England's victory push, and Cook was quick to sing his successor's praises for the way he settled into his new role.
"It was a great week for English cricket," Cook told Sky Sports.
"For Joe Root to get off and win a game by 200 runs and get 190, it doesn't get much better than that. 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."
Cook captained England for a record 59 Tests until calling it a day in February following the 4-0 series defeat in India.
The 32-year-old knows first-hand just how hard a job it is but is confident Root can flourish in the role.
"Nothing can prepare you for it," Cook added. "If you have done a lot of captaincy in county cricket, the role is a very big role, as much off the field as it is on the field.
"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."
Cook, who spent five years as England captain, felt different on the first morning at Lord's knowing his responsibilities were reduced.
But the left-handed opener, England's leading Test run-scorer, is now happy to concentrate solely on his batting.
"It was business as usual for me," said Cook, who scored a second-innings 69 at HQ. "I have played for Essex all year without the captaincy so it hasn't changed.
"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 'its 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.
"The target hasn't really changed for me. It is just to score runs to help set up England victories, that is kind of the role it is. Scoring big hundreds is the dream for any batter.
"As you get slightly older, you kind of appreciate everything a bit more because you know a bloke will always come and tap on your shoulder at some stage in the next few years. I am coming closer to the end of my career than I am the start of it."
Cook is expecting a South Africa backlash at Trent Bridge from Friday as they aim to level the four-match series.
"They will be hurting from that first Test," he added. "They are a quality side.
"You watch a lot of their cricket and they are one of the sides that travel very well. 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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