The Toffees have moved swiftly to replace Stones, who was sold for £47.5million to Manchester City on Tuesday, by acquiring Williams, 31, for a fraction of the cost on a three-year deal.
Williams had spent eight years with the Swans and though they rejected Everton's initial advances for him, the Welshman's desire to move to Merseyside saw the South Wales outfit relent.
Williams spoke of Everton boss and former Barcelona defender Ronald Koeman's standing in the game as one of the key factors behind his decision to switch to Goodison Park.
"I've had eight great years at Swansea and, for me, the new challenge was something I felt I needed, so I'm delighted to have joined Everton," he said on www.evertonfc.co.uk.
"Everton is a great club with an unbelievable manager - a world legend in football - and I believe the club is going in the right direction
This is something I wanted to be a part of.
"I look forward to seeing what the manager's plans are for me and the team
He's been one of the greatest centre-halves to ever play the game, so I look forward to learning from him and seeing what he can teach me."
Koeman's shaping of his squad is expected to continue apace now Stones' departure has given the club a cash windfall.
Sunderland's centre-back Lamine Kone is another target for the Dutchman, though he trained with the Black Cats on Wednesday as he awaits a new contract.
On Williams, Koeman said: "We wanted to bring in a strong player in the central-defence position, a player who knows the Premier League and Ashley fits the bill for us very well.
"He is an experienced defender and the kind of player we need in the team
He is a leader
He showed this at Swansea and he has shown it for his country and he is looking forward to his new challenge here at Everton."
Williams was a key figure in Wales' run to the last four at Euro 2016 earlier this summer and he had been a mainstay in the Swans' defence ever since he joined from Stockport in 2008.
Having risen through the leagues with them, chairman Huw Jenkins expressed his regret that Williams had sought pastures new.
"We are disappointed to lose Ashley
It is definitely not something we wanted to happen," Jenkins added on Swansea's official website.
"But having turned down Everton's initial offer, Ashley expressed his desire for a new challenge at this time of his career.
"Taking into account the excellent service Ashley has given this football club over the last eight years, including as club captain, we reluctantly agreed to his request to join Everton after they made an improved offer.
"It goes without saying that we wish him the very best with his new challenge and thank him for his services."
Source : PA
Source: PA