Spurs faced stiff competition for the Germany winger.
AVB said: "He spoke briefly with the player. All of us wanted to give Lewis a feeling of the club and what he could expect.
"We have high expectations for the player. He has done extremely well. He adds to our squad in different positions and it is a major boost for the deal that we managed to have him in January and it was something that seemed reasonable to do.
Pressed on his good fortune after landing a 22-year-old full international who was running his contract down, Villas-Boas went on: "We were extremely lucky."
"We faced tough competition from all over. But the player was very willing to come to Tottenham.
"He is recognised as an excellent young talent so we're fortunate to have him."