The 24-year-old Brazil midfielder had been linked to Chelsea and Tottenham in recent weeks but instead moved to the Russian side in a deal worth €35million (£30.26m).
Shakhtar are due to take on Borussia Dortmund in the last 16 of the Champions League, while Anzhi are only in the Europa League but Willian insisted it was the right time to move.
He told Globo Esporte: "Of course I thought about everything, I also took the Champions League into account, but it was the right moment to leave.
"In the middle of the year it would make two years I would have been trying to leave without success.
"It's indifferent for me that Shakhtar are in the round of 16 of the Champions League and Anzhi are in the Europa League.
"My objective was to leave from the front door and I think that happened. I don't feel hurt by anybody."
The playmaker is looking forward to teaming up with striker Samuel Eto'o at his new club, saying: "It will be an honour to play with him and to create chances for him."
Willian also talked about past offers, saying: "I want to make it clear that everyone knows Shakhtar turned down many offers from other clubs before, but the value was lower than what Anzhi paid."
Confirming his departure, a statement on the Shakhtar website read: "FC Shakhtar and FC Anzhi reached an agreement regarding Willian's move to the Russian club in a transfer worth 35million euros.
"Shakhtar thanks Willian for the years spent at the club, for his goals and professionalism. We wish our former player success in his new team."
Willian joined Shakhtar from Corinthians in 2007 and played 221 games for the club, scoring 37 goals.