Mauricio Pochettino's injury-hit side looked to be heading for a third successive home defeat in the Premier League after Craig Cathcart's first-half goal, but Son – back from Asian Cup duty – levelled late on and then Llorente made amends after wasting several presentable chances earlier in the match.
Straight back into the team after his period of absence, Son was the first to go close early on, but Watford grew into the contest and went ahead just before the break through Cathcart's scrappy goal.
The wastefulness of Llorente – starting in place of the injured Harry Kane – looked as though it was going to cost Spurs, but Son restored parity and then the Spanish striker completed the turnaround on a dramatic night at Wembley.
Son looked sharp early on and almost opened the scoring in the ninth minute, his 25-yard strike agonisingly missing the top-left corner.
Hugo Lloris did not cover himself in glory in the 38th minute, however, as his positioning was way off at a corner and Cathcart bundled home to beat the stranded Frenchman.
Llorente was guilty of an inexplicable miss at the start of the second half, seeing his initial close-range shot saved, before somehow kneeing the rebound over from about three yards.
Daryl Janmaat almost turned an Christian Eriksen free-kick into his own goal just past the hour, as Ben Foster made a crucial save, before Llorente then headed wide from close in as Toby Alderweireld chipped the ball back into the danger zone.
Tottenham's luck changed in the 80th minute, however, as Llorente nudged the ball on to Son in the area and the South Korean finished emphatically past Foster.
Llorente then finally found his mark three minutes from the end, sending a looping header in from Danny Rose's cross.
What does it mean?
Spurs close on Manchester City It is amazing how much can change in a few minutes. When they were staring at defeat, Spurs were going to be staying just four points above Chelsea and Arsenal. However, with the late turnaround, they now find themselves only two adrift of second-placed City and seven clear of the two clubs behind them. Given the other results this matchday, this could be a potentially crucial moment in Tottenham's season.
Rose a thorn in Watford's side
While it was certainly not a vintage display from Spurs, Rose did catch the eye on the left flank. The full-back got forward constantly, making himself a real outlet, and his cross ultimately provided the winning goal.
