Rasmus Hojlund opened the scoring with his second goal in two league games for the Reds after just three minutes, smashing home past Spurs goalkeeper Guglielmo Vicario.
Ange Postecoglou's side then hit back through Richarlison, with the Brazilian forward rising highest to nod home Pedro Porro's inswinging delivery into the area.
United retook the lead shortly before the interval, with Hojlund and Marcus Rashford combining well, before the latter carefully tucked home to restore our advantage.
The Lilywhites, however, galvanised themselves at the start of the second period and Rodrigo Bentancur smashed home the game's fourth goal, ensuring the spoils were shared.