"There were people who doubted our ability and we were all very keen to show that Spain has a great team and can aspire to everything," the Valencia player told his club's Web site on Friday.
"We were always confident in our ability, we knew what we could do and we will prove that we can do it again."
Villa, the top-scoring Spaniard in the Primera Liga last season with 25 goals, struck twice against Ukraine and has scored four times in nine appearances for Spain.
The 24-year-old, who Valencia bought out of his contract at Real Zaragoza for 12 million euros ($15.20 million) last year, said he was always confident he could produce the goods.
"I knew I'd be able to hold my head up high and perform well and with a bit of luck everything would work out well," he said.
"I might even have scored more goals if I had stayed on the pitch, but the boss wanted to take me off and keep me fresh for the upcoming games and that is also a source of pride to me.
VERY HAPPY
"I'm very happy because they were some of the most important goals of my career, but I hope that there are even more important things still to come."
Barcelona coach Frank Rijkaard said Spain's demolition of Ukraine was no surprise given the quality of their players.
"If you look at the talent of the team, then the result isn't that surprising because Spain have got some very skilful players," the Dutchman told Barca's Web site on Friday.
"The truth is that after the first half it looked as though there was only one team on the pitch. That means that they worked very well together and that is important because it will give the team more confidence and self-esteem."
However, Villa said there was no danger of Spain getting carried away after their comprehensive opening victory.
"When people ring me from Spain they say the fans have gone mad and are delighted with the way we played... But we don't want to get drawn into the euphoria. We know that we have to keep our feet on the ground.
After a day off the players return to training on Friday to prepare for their second match in group H against Tunisia in Stuttgart on Monday. They meet Saudi Arabia four days later.
