After Ander Herrera's red card left the holders with an almost impossible uphill struggle, Kante's sublime 20-yarder early in the second half proved enough for Antonio Conte's side to progress.
United were hit by a striker crisis coming into the contest, with Zlatan Ibrahimovic banned and both Wayne Rooney and Anthony Martial injured.
Marcus Rashford rose from his sickbed to lead the line, and the youngster was arguably United's most influential player.
The Reds fell behind inside a minute in October's league meeting, but this time we started well and soon carved out a decent opening.
Rashford was challenged on the edge of the area and the ball dropped for Henrikh Mkhitaryan who fired narrowly wide from 18 yards.
The hosts then went close themselves as Eden Hazard brilliantly turned away from Chris Smalling and burst through on goal.
David de Gea was equal to the effort, though and turned the ball behind with a marvellous reflex stop.
The keeper again came to his side's rescue again from the resultant corner as, despite being unsighted, he palmed away stand-in skipper Gary Cahill's low shot.
Conte and opposite number Mourinho were kicking every ball on the touchline and had to be pulled apart by the fourth official after a coming together between Marcos Alonso and Phil Jones.
Then came the pivotal moment in the contest on 35 minutes.
Herrera's foul on Hazard was clumsy but the Belgian made the most of it with a vastly exaggerated tumble as he went down like he'd been shot, and referee Michael Oliver dismissed the Spaniard.
That left the Reds with a mountain to climb and Chelsea were dominant from then on.
Victor Moses had an effort blocked, Cesar Azpilicueta went close and then Kante capped another magnificent showing with his decisive goal.
Rashford continued to prove United's inspiration and almost pulled his side level just beyond the hour mark.
David Luiz misjudged a long ball, Rashford ran on to before twisting and turning his way past Cahill, only to be denied by a superb sprawling save from Thibaut Courtois.
Paul Pogba flashed a shot wide and Willian was thwarted by De Gea at the other end but, despite a brave and valiant effort, there was no way back for United.
Overall team performance: 6.5/10
United Man of the Match: Marcus Rashford