The British driver qualified fourth and finished sixth in Sunday's season-opening Australian Grand Prix as the Honda-powered team cast off their image of Formula One under-performers.
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"I'm reasonably happy," said the 24-year-old, who beat both McLarens. "It's a big change from last season that now we're a little bit unhappy with being slower than the Williams whereas last year we were way off.
"A podium is definitely still realistic. I think at the next circuit (in Malaysia) we are going to make a big step forward from here because we will know the tyres a lot more."
BAR switched to Michelin tyres, the same as those used by McLaren and Williams, at the end of last season and Sunday's race was their first with the brand.
"To get three points in the first race is great. There were only two races during last season where I got more than three points and if we can do that consistently we're looking pretty strong," said Button.
"I think we are ahead of McLaren if you look at the lap they stopped on in the race, the first pitstop.
"We are looking strong in qualifying compared to them and also in race trim I would say we seem to be quicker."
BAR, now ahead of McLaren in the constructors' championship, came to Melbourne with a question mark over their form after impressive times in testing.
Some felt that the team, who have yet to win a race since entering the sport in 1999, would flatter to deceive but that did not happen.
"I think we can be a lot closer to the Williams in Malaysia," said Button. "It's a great start for us and we're not jumping around in joy because we want to be quicker, we're not winning yet and we are never going to be happy until we are."
However, McLaren's David Coulthard warned Button not to be under any illusions.
"They were quicker than us this weekend but McLaren in my mind are still one of the two top teams in Formula One, irrespective of the package not working at this race," he said.
"You've seen time and time before where some of the smaller teams on the way up have flashes of brilliance which you can't deny them," he added.
"They've done a good job this weekend. But its a long way between having a strong race weekend and winning the world championship.
"BAR have absolutely delivered what they showed in winter testing. They went backwards in the race because the car is a bit hard on the tyres so you can expect more of the same."
