The result stretched Spurs' strong record against Leeds to six victories in their last seven meetings and provisionally lifted them into second place in the table.
Leeds started brightly in difficult conditions as Sean Longstaff's free-kick, carried by the swirling wind, almost embarrassed Guglielmo Vicario, only for Joe Rodon's header to clip the post.
Tottenham struck first, however, when Mathys Tel, handed a surprise start, justified his selection with a driving run and a deflected effort off Pascal Struijk that wrong-footed Karl Darlow.
The hosts responded and levelled before half-time when Brenden Aaronson's strike forced Vicario into a fumble, allowing Noah Okafor to tap into an empty net.
Spurs nearly retook the lead just before the break as Tel's header struck the crossbar, while Vicario produced a vital stop from Dominic Calvert-Lewin early in the second half.
The decisive moment came shortly before the hour mark. Kudus punished hesitant Leeds defending with a powerful drive from the edge of the area that skipped off the wet turf and beat Darlow at his near post.
Leeds manager Daniel Farke introduced extra attacking options late on, finishing with three strikers on the pitch, but Tottenham stood firm to secure their first league win since early September.
Spurs boss Thomas Frank praised the resilience of his side and the quality of his match-winner. "In these conditions you have to be clinical, and Kudus showed exactly why he is such an important player for us," Frank said.
"It was a tough game, but the lads managed it really well."