Cook, already England's leading Test run-scorer and century-maker, ended the first day 153 not out in a commanding total of 348 for three.
The ex-England captain shared a stand of 248 with current skipper Root (136) on the first day of the inaugural day/night Test on English soil.
Cook, who had already made 193 for Essex in the round of floodlit County Championship matches in June, batted for more than six hours Thursday and rarely looked in any trouble as he compiled his 31st Test century.
He also had the best view of anyone as the the gifted Root made his 13th hundred at this level and set a new England record of fifties in 11 successive Tests.
"He kind of makes it look quite easy. Frustratingly easy," Cook told reporters.
"It's incredible he manages to score like he does. If he's not the best England player I've played with he's right up there.
"His game is phenomenal...He's just phenomenally consistent against world-class bowlers around the world," the left-handed opener added.
The match started several hours later than would usually be the case for a Test in England and Cook told reporters: "It was slightly unusual because you're programmed to play in white kit starting at 11:00am (1000 GMT) with a red ball, it's what we've done for all our careers.
"Suddenly changing it takes a little bit of time. It's just a mental thing.
"I think if you get 150 and you're not out at the end of day you enjoy it," he added.
Cook was not alone in having a fine time, with a typically lively crowd cheering him on in what was the 50th Test played at Edgbaston.
Of their six major home grounds, the Birmingham venue is the one where England have their best record with a win rate of 53 percent.
It was also where Cook made his Test-best score of 294 against India six years ago.
"The crowd enjoyed it and it was a good day for England,
But the 32-year-old warned against rushing to judgement.
"We won't know about the pink ball until two or three years down the line," he said.
"I can't see it not being a success in other parts of the world, whether we need to do it in England is a different matter.
"We have the trouble that it's not that dark until the last hour and doesn't actually feel like a day/nighter until the last hour."
Meanwhile cricket chiefs have been accused of discriminating against the colour blind by deploying the pink ball.
According to the Daily Telegraph, a letter from campaign group Colour Blind Awareness has urged the England and Wales Cricket Board to reconsider their use of the pink ball, which it said was difficult to see for both players and spectators with the condition.
"In the UK there are approximately three million people with colour blindness,” the Telegraph quoted the letter from CBA founder Kathryn Albany-Ward as saying.
"So, it would be extremely difficult to argue that it is reasonable to use a pink ball when so many people are potentially adversely affected," the letter added.
Yorkshire and England batsman Gary Ballance, not playing at Edgbaston, is among those who've struggled to see the pink ball because of his colour blindness, although he recently said he was finding it easier to pick up after the ball had been modified.
And three years ago, Australia opening batsman Chris Rogers withdrew from a pink-ball trial match because he has the condition.
But it is understood all the England players at Edgbaston had eyesight tests in a bid to ensure they could cope with the pink ball.
First Published: August 18, 2017, 8:21 AM IST