Tottenham looked set for a frustrating evening at Wembley when Craig Cathcart headed home a corner in the 38th but Son Heung-min marked his return from the Asian Cup with an equalizer in the 80th.
Llorente then made up for a glaring miss earlier in the game by heading home a cross to secure a second straight late comeback win for Spurs, who netted in injury time to earn a 2-1 win at Fulham in their previous league match.
Mauricio Pochettino's side has had a frustrating week after losing in both the League Cup and FA Cup but this win pulled them within two points of Manchester City, which lost on Tuesday, and built a seven-point gap to fourth place.
Llorente, filling in for the injured Harry Kane, had earlier missed an open net from a couple of yards out when he need the ball over the crossbar on a rebound after Ben Foster blocked his first effort.