The win moves Tottenham within two points of second-placed Manchester City and seven ahead of London rivals Arsenal and Chelsea in the battle for a top-four finish.
Spurs should have been level seven minutes after the break when Llorente managed to put the ball over the bar.
He spurned another chance in the 64th minute when he headed wide.
Danny Rose and Harry Winks also went close for the hosts.
Just as the home fans were losing faith South Korean Son, who has just returned from the Asian Cup, fired the equaliser.
Spaniard Llorente then made amends for his earlier misses by heading the winner to delight the fans who braved the cold.(Reuters/NAN)