Kante rifled home a long-distance effort to see off 10-man United, who had Ander Herrera dismissed in the first half.
The goal secured another miserable Stamford bridge return for Jose Mourinho, with Kante scoring his maiden Chelsea goal in a 4-0 Premier League romp this season.
"I'm happy to score again," Kante told BBC Sport. "I'm happy for the win and for the fans it was a very beautiful win.
"It was difficult, Manchester is a good team, it was a tough game but we won. When Herrera was sent off, it was more easy.
"It's a pleasure to go to the next round, hopefully we can go to the final and see who we are going to play."
After sending Chelsea into the FA Cup semi-finals with the winner against Manchester United, N'Golo Kante reflected on a beautiful win.
Chelsea captain Gary Cahill hailed the performance of Kante and believes the Frenchman has improved further on his title-winning exploits at Leicester City as a Stamford Bridge double looms into view.
"N'Golo has been fantastic," he said. "You saw what he did last season but he has kicked on again this season. He is so important, especially to us at the back."
Cahill said that Chelsea had complained about the treatment dished out to Hazard before Herrera saw red.
"There was a few fouls leading up to that, maybe it was silly to go in and make a foul again," he said.
"We complained that Eden was getting harassed early on. There was many fouls, no dangerous ones."