The Briton went on to finish in second place, some ten seconds behind Vettel.
Button admitted he enjoyed the race, but rued his poor start, as he felt he could have been closer to Vettel.
"It was a really fun race," said Button. "Fighting with Lewis, with Michael, and with Fernando. It was a good race, it wasn't for the lead, which was a shame but it's always very difficult when you are down in seventh.
"I had a tough start to the race but it was a lot of fun, and the atmosphere here is electric. We have to get things together on race weekends and then hopefully we can be challenging Seb."
He added: "I don't know if we could have challenged Seb today. It's a pity we had such a poor start. It is a little issue but it can cost you a lot of time.
"We did a great job with the pitstops. It is great to get a second but you always want a lot more."
Button said passing Alonso for second was a "tense" moment, as he was unable to overtake the Spaniard with the DRS and KERS on the straight.
"It's a weird thing with the DRS. We get to a point on the straight where we don't go anywhere, we sit on the limiter. I made a mistake at Ascari, and Fernando got a gap and I couldn't get any closer, but on the hard tyre Fernando was struggling for grip.
"That is the good thing about KERS: you can catch people out if they make a mistake. It was a tense moment but I eventually pulled a gap and easy to settle into the race."