A fiery opening half saw United start well before midfielder Ander Herrera was sent off to give Chelsea a huge advantage for the remaining 55 minutes.
Chelsea dominated from there and Kante netted the eventual winner from long-range soon after half-time to inflict a second defeat for Jose Mourinho's men this season at Stamford Bridge.
Both clubs provided their lineups ahead of kick-off, with Marcus Rashford starting up front for United despite previously being a doubt because of illness:
United started sharply, looking to press the Chelsea defence high up the pitch, and Henrikh Mkhitaryan dragged an effort just wide from the edge of the box after 12 minutes.
Eden Hazard then burst into life as he turned on the halfway line and drove into the box, only to see his 16th-minute low effort tipped behind by an outstretched David de Gea.
The Spanish goalkeeper then made an even better save from the resulting corner as he got down brilliantly to keep out Gary Cahill's attempt, per Match of the Day's Gary Lineker:
United continued to look impressive going forward but Hazard was causing all kinds of problems when he got on the ball.
And it was two fouls on the Belgian that saw Herrera dismissed for a second booking after just 35 minutes.
It was arguably harsh on United and the Spaniard, with neither yellow card necessarily wholly deserved, per the Associated Press' Rob Harris and AS:
Red Devils manager Mourinho responded immediately by replacing Mkhitaryan with Marouane Fellaini and United were able to reach half-time with the match still goalless, but not before the Portuguese and Antonio Conte clashed on the touchline.
It took only six minutes for Kante to put Chelsea ahead after the break, though, his fine strike from 20 yards wrong-footing De Gea and eluding the despairing Spaniard as the ball nestled in the bottom corner, per BT Sport Football:
It looked a long way back for United given their numerical disadvantage, but just before the hour Rashford brilliantly skipped past Cahill and bore down on goal.
He arguably should have buried the finish to equalise, but Thibaut Courtois deserved credit for staying big and pushing the effort wide, per the Manchester Evening News' James Robson:
Unsurprisingly, Chelsea proceeded to dominate possession and carved out a number of chances to put the game to bed, but Diego Costa spurned two opportunities and Willian forced another decent save from De Gea.
The danger remained that United may find an equaliser from somewhere and Paul Pogba put an effort wide from the edge of the area 10 minutes from time following fine work from Antonio Valencia.
In stoppage-time United then piled forward to try to force extra-time, but Chelsea defended well and eventually claimed victory to knock the FA Cup holders out of the competition.
Mourinho initially said he would not comment on Herrera's red card, but it seemed fairly clear he was unhappy with referee Michael Oliver's decision, per BBC Sport's Tom Rostance:
I don't speak [about the red card]. I just want to say that I'm really proud of my players and Manchester United fans. We have a very important match on Thursday and I want to rest a little bit and prepare as best as we can.
Everybody can analyse from different perspectives but we all watch the match until the red card and after the red card. So we can compare the decisions of the two yellow cards, in this case with others which were not given.
I don't want to go in that direction. Michael Oliver is a referee with fantastic potential but in four matches he has given three penalties and a red card. I cannot change that. I shook his hand and said many congratulations.
Meanwhile, Conte credited United and said he was happy to see his team through to the next round, per Rostance: "It was a good performance against a strong team with good players. United has the best squad in the league. For us we must be pleased to go into the next round."