The pressure has been mounting on Silva following a disappointing start to the Premier League season and defeat against the Owls would have given him an even bigger headache.
However, the Toffees put their shocking away form behind them as two goals by former Sheffield United-man Dominic Calvert-Lewin in the opening 10 minutes, booked their place in the fourth round of the competition.
Silva was delighted with the win and the performance of three of the players he brought into the side, Calvert-Lewin, Alex Iwobi and Djibril Sidibe.
Despite the victory, the Portuguese will not rest on his laurels and has quickly turned his focus to the weekend when the Toffees entertain Premier League champions Manchester City at Goodison Park.
Whether Iwobi and co have done enough to keep their place, remains to be seen, but they have done their chances no harm.
"We reacted well tonight and now we move on to Saturday," Silva said.
"When we signed [Alex] Iwobi we knew he could play behind the striker and tonight was a good moment to see him there. He can play in both positions and do well in both."
Silva also pointed out the importance of the competition to Everton, and reiterated they will be going all out to go as far as they can in it.
Striker Calvert-Lewin could have a key role to play in the competition following the summer arrival of Moise Kean from Juventus and the boss says it was good to see him get a brace under his belt.
He added: "It was important for Dominic [Calvert-Lewin] to score twice. He did well to score in important moments. It gives him confidence. Let's hope he continues in that way."