Going into the game against AC Milan two goals behind from the first leg, the Spanish side cruised to victory scoring four times on the night, with Villa, Jordi Alba and Lionel Messi getting on the scoresheet.
Villa told reporters: "The fans were always behind us, they always trusted us. They knew we could do it and in the end we did it for them.
"When you go through on a night like this, all of the bad days are forgotten. After experiencing what we did tonight, we don't care who we get in the quarter-finals."
Barcelona will find out their opponents in the last eight when the Champions League draw takes place on Friday.