A quite remarkable contest saw the hosts fight back from going 2-0 down in the 64th minute, with Leighton Baines scoring what proved the winner with a penalty in stoppage time and Watford's ex-Everton midfielder Tom Cleverley then sending a last-gasp spot-kick wide.
The dramatic events during 13 minutes of time added on at the end came after goals from Richarlison, in the first minute of the second half, and Christian Kabasele, were cancelled out by efforts by Oumar Niasse and substitute Dominic Calvert-Lewin.
Silva - who has been linked with the Everton job, along with the likes of Sam Allardyce and Sean Dyche - said after his ninth-placed side's third successive defeat: "This is really disappointing, for us and our fans.
"I think we did everything well until it was 2-1. We controlled until that moment. In that moment we gave one chance for the Everton players to believe again, and we started to be a little bit uncomfortable."