Malky Mackay said: "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 tonight who have game-changing players.
"They had two centre forwards, one who has cost an awful lot of money and another at some point who will cost a lot of money.
"We have to give due respect to the team we played. 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."
In-form forward Mason, who opened his Bluebirds account against Barnsley last weekend, proved the difference as his second goal in four days after 41 minutes settled this all-Championship tie.
The Irishman cost just £250,000 from Plymouth in the summer and the talented youngster showed a maturity beyond his 20 years to book a place in the last eight.
Mackay said: "It was a mature finish from Joe. His movement and game intelligence belied his experience. It was a beautiful goal.
"The boy has great potential but we have to work hard to keep nurturing him.
"He has good people in terms of players and coaches around him and he's level headed. If he keeps working hard he will only get better."
Mason almost broke the deadlock in the sixth minute when he was set free by Craig Conway, but saw his shot clip the outside of Lee Grant's left-hand post.
However, the hosts had goalkeeper Tom Heaton to thank for remaining level moments later after the ex-Manchester United trainee produced a fine double stop.
Heaton reacted smartly to touch Keith Treacy's fizzing drive onto the post before Cardiff's stand-in stopper acrobatically tipped Brian Easton's header over the bar from the resulting corner.
But Cardiff held the upper hand and Mackay's men finally made their dominance count through Mason's opener in the 41st minute.
Cowie surged forward in midfield unattended and fed Mason on the edge of the box, who produced three lovely touches to control the ball, set himself and then coolly finish left-footed beyond Grant.
Burnley manager Eddie Howe said: "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."