Goals from in-form Gareth Bale, Gylfi Sigurdsson and Jan Vertonghen secured a vital victory to take to Italy for the return leg next week with Spurs in charge for a large portion of the game.
Inter missed a good chance in both halves but were underwhelming for most of the game and Villas-Boas believes his side were deserving of the result.
"It was a very good performance and a great night of European football for us," he said.
"We really felt that at half-time we could have gone further - there were some important moments of the game for Inter Milan too but we had so many opportunities - we really deserved to win.
"Unfortunately for us it doesn't mean we are through yet, it is a difficult place to go and I just hope we can do enough to qualify.
"I think the result is what it is and it is well deserving for Tottenham and still prestigious."
The Spurs boss refused to criticise Bale who was shown a yellow card for simulation and will miss the return leg in the San Siro through suspension.
Villas-Boas defended his star player and said that, following the win, he may have even considered asking the Wales international to collect a booking on purpose to clear his suspension out of the way.
"It is a difficult decision for the ref," he said.
"I won't build on that case. I don't want to go into that, we have a wonderful player to enjoy and it is ridiculous if we go on like that.
"He is obviously a great miss but to be fair, probably with this result we would have asked him to get the yellow card to be clean and ready for the last eight.
"We are in a good position but it is not finished so we will miss Gareth because he has been influential in consecutive games but hopefully we can make it without him.
"He is a tremendous player so obviously he is going to be missed but an opportunity might arise for another player."
Inter coach Andrea Stramaccioni conceded his side had lost to the better team on the night but maintained they will still fight to secure a quarter-final berth.
"Obviously I believe that they played better than us and deserved what they got," he said.
"First of all we suffered from a physical point of view, they created us lots of problems with their aerial game but having said that, even if we had a negative night we had two great chances to score and we have not been able to do it.
"We all know that it is going to be very hard, a tough situation to face but at the end of the day we are Inter.
"We want to win the second leg and, as we all know, football is a beautiful game and you never know what will happen but we are fully aware it is going to be very hard."