Riyad Mahrez put the Foxes ahead after 37 minutes with his sixth league goal of the season but Malian defender Molla Wague, making his first Premier League start on loan at Watford from Udinese, equalised just before the interval.
Abdoulaye Doucoure, returning after a one-match suspension, won it after 65 minutes with a shot that went in off Leicester goalkeeper Kasper Schmeichel.
“Commitment, attitude and spirit are the most important things, more important than everything. After going 1-0 down we showed all these things,” said Silva.
“We have shown it from the first moments of the season. We didn’t perform like we should have the last two games. But it was a very good game against a good team.”
It looked unlikely to turn out that way when Mahrez nodded home Marc Albrighton’s left-wing cross in the 37th minute.
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Richarlison might have put Watford level almost immediately when he took Andre Carrillo’s pass and went round Schmeichel, but the angle was too narrow, his shot hit a post and it seemed Watford were heading for defeat again.
But when Leicester failed to clear a corner in the final minute of the first half, Wague ran onto a rebound off Wes Morgan to hit the ball home from a couple of yards out.
After 65 minutes Schmeichel was rooted to his line as Tom Cleverley’s free kick found Doucoure beyond the far post. The midfield player’s shot might have been going in anyway, but Schmeichel’s attempt to save made sure that it did.
After Cleverley had missed a good chance to seal the points, robbing Ben Chilwell before shooting past the far post, Watford goalkeeper and acting captain Heurelho Gomes kept his side ahead in a frantic final five minutes with late saves from Morgan and substitute Leo Ulloa.
“Heurelho performed really well and he is a leader in our dressing room,” said Silva.
Leicester manager Claude Puel was disappointed that Watford’s goals had followed a corner and a free kick. Leicester have now conceded more goals – 13 – than any other Premier League sides from set plays.
“Until this afternoon we played with good energy and positivity,” said Puel. “But we have to correct our defending from set pieces.”
Watford (4-2-3-1): Gomes 7; Janmaat 6, Wague 6, Kabasele 6, Zeegelar 6; Doucoure 7, Watson 6 (Prodl 86); Carrillo 8 (Sinclair 89), Cleverley 7, Pereyra 5 (Okaka 44, 5); Richarlison 6. Booked: Watson, Kabasele, Zeegelar. Goal: Wague 45, Schmeichel og 65 .NEXT UP: Swansea (h), Sat PL.
Leicester (4-1-4-1): Schmeichel 5; Dragovic 5 (Gray 73), Morgan 6, Maguire 5, Chilwell 6; Mahrez 7, Ndidi 6, King 6 (Ulloa 84), Albrighton 7; Okazaki 6 (Slimani 73, 5); Vardy 5. Booked: Maguire, Dragovic, King. Goal: Mahrez 37. NEXT UP: Liverpool (a), Sat PL.
Referee: C Kavanagh (Manchester).