Ander Herrera was sent off in the first half after a pair of bookings for fouls on Eden Hazard, giving Kante the space he needed to score early in the second half.
United had David De Gea to thank for keeping the score level early on as he tipped an Eden Hazard shot wide after the Belgian danced through the box.
De Gea was then called into action again to stop Gary Cahill’s half-volley following the corner.
Hazard was the danger man, and Herrera received his first yellow card for bumping him off the ball on 20 minutes.
And a second booking for Herrera followed 15 minutes later when he tripped up Hazard from behind.
Jose Mourinho then removed Henrikh Mkhitaryan in favour of Maraoune Fellaini, who had been in line to start before Marcus Rashford made a late recovery from illness.
The Man United manager then got into a shouting match with counterpart Antonio Conte after Antonio Valencia’s challenge on Marcos Alonso.
United held on with 10 men until six minutes after half-time, when Kante got lose to score from about 22 yards.
Running on the ball to his right atop the box, Kante caught De Gea leaning the wrong way and curled his shot to the goalkeeper’s left to open the scoring.
United’s best chance at an equaliser came just before the hour mark when Rashford spun away from David Luiz at the halfway line, then turned Cahill the wrong way as he ran in on goal, only to see Thibaut Courtois save his shot.