Silva had been heavily linked with Everton before Allardyce was hired in November, having enjoyed a fine start to the season at Watford.
The Hornets signed Silva at the beginning of the campaign after he impressed at Hull City, despite their relegation at the end of the 2016-17 season.
But they cited Everton's initial approach as a major catalyst for Silva's sacking in January, following a run of one win in 11 Premier League matches.
When Watford refused to give Everton permission to speak to Silva, they turned to Allardyce and he ultimately steered them clear of trouble, finishing eighth in the Premier League.
But supporters never took to the former England manager and Everton confirmed his exit on May 16, paving the way for Silva – who was always the big favourite – to make his return to Premier League football.
And the Portuguese coach says he is aware of the fans' expectations after Allardyce failed to truly convince them.
"I'm really proud to be the new Everton manager," he told the club's official website. "I'm excited and I'm really happy to take this big challenge for us as a club and for me as a manager.
"I want to express my thanks to Farhad [Moshiri, majority shareholder], to the chairman, to the board of directors and to Marcel [Brands, director of football] as well for the confidence in my work.
"I know the huge history of Everton as a club and what the fans expect. I'm sure with everybody working together I will be ready for this challenge.
"We have a lot of work to do in this next month and it is important to prepare everything in the right way. But there will be even more important work to do from the start of pre-season into the first Premier League match of next season.
"I know what our fans expect - they expect results but not only results. I want our fans to be proud when they see our team on the pitch. I want them to feel that we are committed, that we are working hard and enjoying our football because that is important as well.
"Everton is a really ambitious club and that is what I want. What we are seeing now are good changes at the club. The club is changing its approach, but one thing we cannot change and nobody wants to change is the huge history and ambition of the club."