An early goal from Eric Dier proved to be the difference and although Cardiff had Joe Ralls dismissed with just over half an hour remaining, the Bluebirds will count themselves unfortunate to miss out on a point after a much-improved performance.
However, Cardiff now sit bottom of the standings due to Huddersfield Town's draw at Burnley, while Neil Warnock's team are now eight games without a win.
Spurs started well in North London and it took the home side just eight minutes to go ahead as Dier converted from close range after the ball had fallen invitingly to him following an inswinging cross from the left.
Cardiff almost delivered an instant response as Harry Arter had a low shot comfortably saved by Hugo Lloris, but Spurs soon went in search of a second and Davinson Sanchez was denied from 12 yards out by a solid stop from Neil Etheridge.
The home side remained the better team but Cardiff came within inches of getting back on level terms midway through the first half, with Toby Alderweireld having to clear off the line after Josh Murphy had clipped a shot over the advancing Lloris.
Sol Bamba then headed narrowly over, and it led to Spurs needing to up their concentration having only given the visitors encouragement that they could net just their fifth goal of the top-flight campaign.
Son Heung-min teed up Moura who failed to hit the target from eight yards, before the South Korean advanced into the penalty area only to blaze high over the crossbar.
Spurs came agonisingly close yet again within 60 seconds of the restart, with Kane seeing a header well saved by Etheridge before the rebound from Moura was somehow cleared off the line by Sean Morrison.
Cardiff responded well but just before the hour mark, the Welsh outfit were reduced to 10 men when Ralls was handed a straight red card for cynically taking out Moura as the Brazilian threatened to run clear of him on the right.
The visitors disputed the dismissal but despite being a man light, an equaliser almost arrived soon afterwards with Lloris having to tip a Morrison header onto the post.
Spurs remained in the ascendancy but they were unable to create anything significant in the final third with Cardiff keeping everyone behind the ball and looking to launch counter-attacks.
The remainder of the game followed a similar pattern with neither side being able to get any shots on goal, but that suited Spurs and Pochettino who have got themselves back into contention at the top of the table.
TOTTENHAM HOTSPUR(4-2-3-1): Lloris; Trippier, Alderweireld, Sanchez, Rose (Davies 88'); Dier, Winks; Sissoko (Wanyama 85'), Moura, Son (Lamela 72'); Kane
CARDIFF CITY (4-3-3): Etheridge; Ecuele Manga, Morrison, Bamba, Bennett; Camarasa, Arter, Ralls; Paterson (Ward 78, Reid 85'), Murphy, Hoilett (Harris 78')