Speaking after Spurs' 1-0 defeat to Premier League rivals Arsenal, which was sealed by Olivier Giroud's strike midway through the first half, Villas-Boas was quick to stress that the club would move on despite Bale's departure.
Bale - who signed a six-year deal at Real after joining on an undisclosed fee - was a key figure for Tottenham in the 2012-13 season, scoring 26 goals in all competitions, and Villas-Boas insisted that the Welshman will be fondly remembered at White Hart Lane.
"We wish him all the best, he's left us with great memories from last season, which I think every Tottenham fan cherishes," Villas-Boas said.
"But he has moved on and we move on as well."
Tottenham have recruited heavily in anticipation of Bale's move to the Spanish capital - bringing in the likes of Roberto Soldado, Erik Lamela and Christian Eriksen – but they were unable to break down a dogged Arsenal side.
Jermain Defoe was denied an equaliser as Wojciech Szczesny produced a stunning save to keep out the England international's volley late in the second half, as Arsenal held on for victory.
Despite his side's loss, Villas-Boas was still satisfied with their display.
"I think (in) the first half they had good chances. It was always a competitive game with chances for both teams," he told SkySports.
"We kept on trying to get the equaliser, drawing was never going to be enough, but they had that clinical ability to score.
"(There was) great spirit from the players to approach the game in such a courageous way, to expose Arsenal in the second half."
Etienne Capoue, a recent recruit from Toulouse, was taken off on a stretcher with an ankle injury, and Villas-Boas said that the Frenchman's absence may be a lengthy one.
"He is being x-rayed now, it doesn't look too good," he added.
