Villas-Boas was particularly happy with his side's first-half performance and, although he admitted that they lost a degree of control over the match in the second period, felt that they were worthy recipients of all three points.
"It was a very good first half, where we were in control. It is always tricky because Anzhi looked to be very threatening on the counter-attack, so attacking our spaces whenever we lost the ball," Villas-Boas told PA.
"But I think through what we did in the first half we deserved the result. We lost a little bit of control in the second half, kept giving the ball away and they had the upper hand, but even so we were defensively sound."
The win leaves Tottenham four points clear at the top of Group K after just two games.