South Africa were looking solid at 140-2 at The Oval on Sunday before captain De Villiers was caught a yard out of his ground, with Miller exiting in the next over after a disastrous mix-up with Faf du Plessis.
India took the last eight South Africa wickets for 51 runs and a serene Kohli was unbeaten on 76 as his team cruised home with 12 overs to spare to set up a likely semi-final with neighbours Bangladesh at Edgbaston on Thursday.
Kohli felt the self-inflicted dismissals of De Villiers and Miller - with Imran Tahir also run out to conclude the innings - contributed heavily to India's crushing win.
"Their batsmen looked pretty confident," Kohli told reporters at his post-match media conference. "If you get two run outs pretty quickly, then the mindset totally changes.
"As a captain, you have to understand where the game is heading after that. We only wanted to go for wickets because we understood now they are probably hesitating in going for their shots. We got them attacking fielders and we asked the bowlers to make them play difficult shots.
"I think initially before AB got run out, he did really well and Faf did really well. I think his run out could be the turning point, and David Miller's; he's a very dangerous player as well. They can drag the team up to 260, 270 from any sort of situations. I think those two run outs were the turning point today."
Happy times after winning d match!! Rabb da shukrana ada karte hue!! Rabb rakha @imVkohli @y_umesh pic.twitter.com/oqI09FoHL9
— Shikhar Dhawan (@SDhawan25) June 11, 2017
After Sri Lanka chased down 322 in India's last outing, Kohli highlighted the improvement in his bowling attack to restrict South Africa to a below-par score.
"I'm really delighted for all the bowlers to have executed all their plans well in such a big game," Kohli added.
"To be able to bowl like that against a batting line-up of their capability is something that already we can feel that confidence and that energy in the changing room because of a bowling performance like this, and we surely would like to take that forward into the semis.
"The bowlers stuck to their lines and got us the breakthroughs when we required and that really helps when you chase that kind of a total, now you can take your time because the bowlers have done such a good job."