Writing in The Times, Carter said the trio, who play for the All Blacks, Ireland and England respectively, ran their respective teams but each had different styles of play.
Barrett had X factor whether he was playing at first five-eighths or fullback and his lightning speed allowed him to do freakish things that were beyond Sexton or Farrell.
"As a flyhalf you need to be demanding of the players around you, which comes naturally to Johnny and Owen. Beauden has had to grow into that role over the past couple of years. It is not as natural for him as for the other two, but flyhalves need to be able to control games," Carter said.
In coping with pressure there had been questions over Barrett but he was a class player with a cool head who read the game well and he would learn from the setbacks he had suffered. These included missing goal kicks, especially in the series against the British & Irish Lions in 2017.
That was the closest thing to a World Cup, Carter said, and he believed that Barrett would have learned plenty from the Lions series.
When it came to kicking under pressure Farrell was right up there.
"He may have had the odd bad game kicking-wise, but you just have to look at his career. He is accurate and professional so there are no doubts at all, come the big occasion, that he will step up," Carter said.
In physicality Farrell was a lot different to Barrett.
"When you see your No.10 putting in a big shot, it inspires the people around him. He got a couple wrong last season, on André Esterhuizen and Izack Rodda, but his temperament has changed. Even though he is a physical player, he is a lot more level-headed than a few years ago.
"Johnny, likewise, is brave. He puts his body on the line and gets targeted a lot in games. He runs the team extremely well. He probably doesn't have the attacking flair of Beauden but is very good at game management," Carter said.
Leadership was a requirement in the first five-eighths position and each of the trio were grounded, down-to-earth, hard-working players.
When it came to decision-making Carter said he was successful because he backed his instincts.
"It was not because I was a robot and did what the coaches said and stuck to the game plan at all costs.
"If an opportunity presents itself, you need the players to be able to express themselves and to make the most of it. I think these players have been around long enough that they back their skills and their decision-making to be able to take those opportunities that may arise in a Test match and deviate slightly from the game plan," he said.
When it came to the key asset of kicking from hand, it was important that five-eighths keep rush defences guessing.
"That is a key part of shaping a defence and it is done well on occasions by these players. Johnny is superb at picking the right kick for the right moment. Jonny Wilkinson was the master at it with his left foot or right foot, and it was bang on the money every time. He was the one who re-established that part of the game and it is now a big part of all those fly halves' games," he said.