The Spaniards had taken the lead through Nico Williams early in the second half but a Cole Palmer equaliser left them on tenterhooks until Oyarzabal's late strike.
"Euphoric. All our people deserve it, we suffered a lot," Williams said. "They had a good team, they have players who can make the difference but we were able to counter their weapons and in the end we won the European Championship and we're very happy."
His fellow winger Lamine Yamal followed up his 17th birthday on Saturday with being crowned European champion a day later.
"Very happy, and looking forward to coming back to Spain and celebrating with all the fans. It is really the best present there is," he said.
"It's a dream. When we were drawing it was difficult, they pushed us hard. I don't know what this team is made of, but we always come back, we always win."
A weak shot from striker Oyarzabal had preceded England's equaliser in the 73rd minute, but he made up for it with a brilliant sliding finish to dash English hopes late on.
"I did what I had to do at every moment to try to help," he said. "I was lucky enough to give us the victory. To have the moment to help as it has happened to me is the best. It happened to me, but could have happened to anyone."
Spain coach Luis de la Fuente said he was delighted to see his team confirm their status as one of the world's best.
"I couldn't be happier. To see the fans, to see the players. A real team, European champions. I said I was proud and today I am even prouder. It confirms what we are. For me, they are the best in the world and today I confirmed that definition," he said.
The Spain coach also sounded an ominous note of caution for any side seeking to knock the Spaniards off their perch.
![Spain players celebrate victory Spain players celebrate victory](https://livesport-ott-images.ssl.cdn.cra.cz/r900xfq60/5dc9d705-d7e6-4b0b-9f19-eccb47f29add.jpg)
"There is always room for improvement - the virtue of this team is that we think we can be a little bit better," he said.