England were in the driver's seat for much of the match but Australia's often cautious approach prevailed thanks to a match-winning 55-run ninth-wicket partnership between captain Pat Cummins and Nathan Lyon.
It was England's third defeat in 14 tests since McCullum joined captain Ben Stokes at the helm last May.
"Obviously you'd rather have won the game, but the way that we played, I think it's validated our style," McCullum told British media. "If we'd have got a little bit of the rub of the green then we might have been on the other side of it.
"It was a cracking test match and two very different styles of play. But like a heavyweight boxing match, not everyone has to fight the same.
"I'm sure they (Australia) will stick solid to that strategy all the way through, which I think is great because we'll go a little harder. I think it makes for a really entertaining next few test matches."
Stokes said after the match he did not regret declaring England's first innings on 393-8 on the first day and McCullum backed the skipper's desire to take the game to the opposition.
"We always want to try and take the game forward," he said. "We want to try and seize opportunities where we think we can put opposition teams under pressure.
"We firmly believe, the skipper and I, that this gives us our greatest chance. I would be very surprised if there was too many people who disagree with how we go about playing, because everyone was left entertained.
"You're not always going to win and we understand that. We want to keep getting up and throwing punches as a team, and I'm really proud of the way the guys played."
The second test of the five-match series begins on June 28 at Lord's.
