New Zealand's batsmen scored over 700 runs and their bowlers took all 20 wickets but England completed a 124-run win in the final 10 overs of the fifth day on Monday courtesy of the all-round brilliance of New Zealand-born Ben Stokes.
The Black Caps' attacking mindset drove them to a maiden World Cup final in March and McCullum said his team would not alter their style for the second and final test at Headingley starting on Friday.
"It hurts a lot. I certainly won't lie. But at the same time there is an element of pride that we continue to play the style of cricket that we know gives us our greatest chance," McCullum told reporters.
"There are times we know that teams are going to be able to stand up to you and withstand the pressure you apply to them and come out on top.
"When that happens you've just go to doff your cap and say well played to them and make sure that next time you get that opportunity you go hard again and make sure that you're asking the same sort of questions.
"Who knows at Headingley ... we may see some different results but I'm pretty sure our guys will continue to play the same style of cricket which we have in this test match and which we have done over the last 18 months."
McCullum said being defensive had not worked for New Zealand, who have not lost a test series home or away since being swept 2-0 on their last England tour in 2013.
"For a long time we had that kind of mentality and it didn't serve us anywhere near as well as the aggressive mindset that we are now taking into games," the 33-year-old said.
"You've got to make sure you're authentic to that style, under pressure or whether you're on top.
"We scored over 720 runs in this test match. We took 20 wickets. We ticked off a lot of things which normally would lead to a test win but in some of the key moments we weren't quite able to come out on top.
"The message to the team will be that we weren't far away.
"There's certainly no knee-jerk reaction to a performance like this. For the most of it, we were pretty good."
(Writing by Ian Ransom in Melbourne; Editing by Julian Linden)
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