Morgan became the first England men's player to reach the milestone - Danni Wyatt and Jenny Gunn had already done so for England Women - and the fourth man overall after Shoaib Malik, Rohit Sharma and Ross Taylor.
Morgan, who led England to victory in the 50-over World Cup in 2019 and now has his sights set on the T20 World Cup in India in October, was applauded by his team in a huddle after a short presentation by Buttler before the start of play.
"It nearly brought me to tears to be honest," Morgan told the host broadcaster after England romped to victory with 10 balls to spare, thanks to a seriously quick display from Mark Wood and a stunning 83 not out off just 52 balls from Buttler.
"It meant a huge amount, He's a great friend of mine, our families are quite close, and playing in 100 games is a proud moment for me. But just the words that he said, really did sort of warm my heart and I'm very grateful for it."
Morgan was pleased with the start his quicks, Wood, Jofra Archer and Chris Jordan, gave the side after a solid first over from spinner Adil Rashid, opening the bowling for the second time in three matches this series.
"The carry that we saw throughout the whole game was a bit of a surprise to us, " he said. "Our guys don't find that too hard to adjust to and enjoy it when there's a bit of pace and bounce, so I thought the first half of our bowling was exceptional. When you look at the lines and lengths that we bowled, the plans that we had, and obviously taking early wickets as well in such a long batting line-up, makes a huge difference in the game.
"Today, probably of all the games, the wicket stayed very similar, but I do think there is another challenge with setting a total, particularly when there's one short side on the ground, you just don't know what a good score is going to be. And I think if you lose early wickets on top of that, that raises another question, Obviously you have to play well to be in the game, and then take advantage of it.
"It's just the dynamics of T20 cricket, it just goes to show how strong every side in the world has to be and the preparation that goes into every series, obviously with the World Cup in seven months' time."
And he spoke highly of Buttler, who took on spinner Yuzvendra Chahal with two sixes in his first over and two fours in his second, helping himself to 14 runs off Shardul Thakur in between as he moved to 43 off just 17 balls by the end of the powerplay.
"Jos is a world class player," Morgan added. "We're very lucky to have him. Since he's taken on the role at the top of the order, he's averaged upwards of 50 and striking at 150, so he competes with the best in the world when it comes to opening the batting. He's been in incredible form, he's a great leader within our group. He's been around for an incredibly long time, and in the change-room obviously he has a huge influence."
Buttler, who shared an unbroken 77-run partnership with Jonny Bairstow, hailed the win as "a great team performance".
"Any time you take early wickets in a powerplay really sets you up, and we've got some good partnerships together to see us home," Buttler said.
"In a little way sometimes people don't expect me to attack the spin as well so I thought I'd like to try and take it on. Chahal's a fantastic bowler, had some good battles against him. But it's nice to chance my arm a bit today."
As for what he said to Morgan before the match, Buttler added: "I tried to get across what a special achievement it is and what he's done for English white-ball cricket.
"Today is about him and reaching that milestone. He's been a pioneer for England white-ball cricketers. As a leader he has taken English white-ball cricket to a place it's never been before.
"He's always been ahead of the curve, alongside Kevin Pietersen. They were the two guys that I always looked to in white-ball teams who could do it a bit differently."
Wood finished his first two-over spell with 2 for 10, as India limped to 24 for 3 in the powerplay, and ended with match figures of 3 for 31 off four, after Virat Kohli's unbeaten 77 had lifted his side to 156 for 6.
"I was just trying to bang into the wicket on a good length as fast as I can, trying to make something happen," Wood said. "He [Kohli] is a fantastic player, very hard to bowl at. Even some of my plans at the end, he played great shots off those and sometimes you have to say, 'well played'."
Valkerie Baynes is a general editor at ESPNcricinfo