A moment of quality from Joe Mason sent Cardiff City into the quarter-finals of the Carling Cup for the first time since 1965.
The 20-year-old curled a left-foot shot into the bottom corner from just inside Burnley's penalty area on 40 minutes.
Burnley's best moments came early in the first half when Keith Treacy and Ben Mee tested home keeper Tom Heaton.
Mark Hudson could have made the final 15 minutes more comfortable for Cardiff but he headed over from two yards.
Burnley, who arrived in the Welsh capital having won the three previous League Cup meetings between the sides, managed to push Cardiff back deeper into their own half in the closing stages but failed to carve out a clear opening to draw level.
Cardiff were the better team in possession, with Don Cowie, Craig Conway and Mason linking well in a 4-3-3 formation.
Mason, a £250,000 summer signing from Plymouth Argyle, wasted the easiest chance of the match when he grazed the outside of the post from six yards after just four minutes.
It gave early notice of the young striker's intelligent movement off the ball, which unsettled the visiting defence throughout.
Burnley's purple patch came in the opening 20 minutes when they looked dangerous on the counter-attack.
Treacy forced Heaton into action to push his low effort around the post at full-stretch, and the stand-in keeper turned Mee's header onto the bar from the resulting corner.
But Cardiff delivered the decisive blow just before the break when Cowie's ball found Mason in space in the penalty area and the striker side-footed the ball expertly past Lee Grant's despairing dive.
Burnley finally began to exert a degree of pressure in the final 20 minutes, but defender Mee wasted the only real chance when he sent a free header wide, while Heaton saved comfortably from a Jay Rodriguez shot.
Bluebirds captain Hudson then headed over after Grant misjudged Peter Whittingham's cross, but the miss proved immaterial as Cardiff held on for victory.
Cardiff City manager Malky Mackay:
"A 1-0 win, a clean sheet, no injuries and through to the quarter-finals of the Carling Cup is a good night.
"We were playing against a good Championship team who have game-changing players.
"I was delighted with the way we passed the ball, we created three great chances in the first half and didn't allow them a shot in the second half."
Burnley manager Eddie Howe:
"It's disappointing to lose the game. We started the game really well and tested their goalkeeper early on.
"It was a positive start and on another day we could have scored a couple of goals.
"We also finished the game well and applied a lot of pressure with good possession, but we didn't work their keeper enough or hurt them in the second half."