League One Bluebirds reached the fourth round of the competition for the first time since their run to the final in 2012, thanks to goals from Joel Colwill and Callum Robinson.
Zian Flemming pulled one back for the Clarets, but the Bluebirds held firm to claim a famous scalp.
For Barry-Murphy, the manner of the performance was just as important as the result.
The Cardiff boss praised his players for refusing to retreat into a defensive shell and instead taking the game to their top-flight hosts.
“It means a lot to us to play the way we did and bounce back from what was undoubtedly a disappointing day for us on Saturday,” he said.
"It means a lot in terms of how we want to play. I think when you come to play a team in the Premier League, you've got two choices: you can either step back and hope for the best or try and take your destiny into your own hands.
“We were very aggressive in the way we played the game, and the lads were on the front foot right from the first whistle.”
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Burnley manager Scott Parker made sweeping changes to his line-up, naming an entirely different starting XI for the second cup tie in succession.
The decision contributed to a disjointed opening half-hour, in which Cardiff grew in confidence.
Joel Colwill opened the scoring on the half-hour mark after a corner routine led to a scramble inside the box.
Perry Ng recycled possession, Will Fish nodded into Calum Chambers’ path, and the Cardiff skipper squared for Colwill to fire in from close range.
Robinson doubled the advantage just five minutes later, finishing off a clever move initiated by Rubin Colwill’s cross.
The two-goal cushion was no more than Cardiff’s endeavour deserved, with Burnley creating little aside from a wasted chance for Armando Broja before the interval.
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Parker’s men were sharper after the restart, pressing higher and seeing more of the ball.
Their persistence was rewarded when goalkeeper Nathan Trott gifted possession to Marcus Edwards, who teed up Flemming to curl a superb 56th-minute strike into the top corner.
That lifted the home crowd and set up a nervy final half-hour, but Cardiff’s discipline, organisation and sheer willpower ensured they protected their lead until the final whistle.
Barry-Murphy stressed that his players’ willingness to take risks defined the win as much as their defensive resilience in the closing stages.
“I think you have to show character all the time in sport as in life,” he said.
“There will have been question marks in the players' minds, as there would have been in our supporters' minds.
“But for the supporters to come here on a Tuesday night, this length of a journey and probably do the same distance on Saturday [for the League One trip to Wigan Athletic], is testament to them. We wanted to reward that and give them a performance to enjoy.”
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