Former Barca striker Villa congratulated his old team-mates on an "amazing" victory after they thrashed PSG 6-1 in the second leg of their last-16 tie, overturning a four-goal deficit.
Three of their goals came from the 88th minute onwards and sealed the biggest comeback ever seen in the competition.
And Villa thinks it is the latest triumph from a team that compares favourably with any other from the game's past.
"I think it is if not the best, then one of the best teams in the world - not only in Barcelona history, in all the teams," said the New York City FC star.
"They have won a lot of titles, a lot of games. They make games, crazy games, like this, it is so difficult in 2017, in soccer, in a competition like the Champions League.
"It is one of the best games in the history of the Champions League, it is the best comeback.
"I stayed there three years, I know what happens in this club, I know how great this club and this team is and they deserve it because they made a very great, great, game.
"It is not easy, the other team is PSG, a great, an amazing team. I believe always in Barcelona because they can do this. I'm so happy, I'm so happy for them.
"We couldn't watch this amazing game because we are training in the gym that day, but we finished in the gym and watched the highlights.
"It was amazing for everybody who loves soccer, for me more than the other people as I am a Barcelona fan - I played there."
Villa heaped praise on Neymar after he produced two goals and an assist in the frantic finish to the match, while also hailing the focus of outgoing head coach Luis Enrique.
"Neymar is great for Barcelona," he said. "This guy is great. He is young and has done amazing things because he scored a lot of goals in Brazil, and now in Europe.
"He is the best one by one, he is always happy, he is always help to the team, in goals, in assists. Supporters of Barcelona like me are thankful we have him in our team.
"[Luis Enrique] was my idol when I was a kid, when he was playing with Barcelona - now he is coach.
"I admire a lot his personality as a player and a coach, I only am sure is that he is 100 per cent focused on finishing well this year, not thinking about the next step."