The youngster netted in the Bluebirds' 5-3 win over Barnsley on Saturday and he bagged the only goal of the game in the 39th minute to propel Malky Mackay's men into the quarter-finals.
Mason opened the scoring on 39 minutes, curling home from the edge of the area to hand the Bluebirds the lead. The former Plymouth man had earlier hit the post from just inside the box, while Craig Conway and Don Cowie should both have done better as Cardiff enjoyed the better of the first-half chances.
Mason had a goal disallowed for offside after half time and, while Ben Mee headed over and Dean Marney's free-kick went close for the visitors, Burnley could not find an equaliser.
After Marney brought a simple save out of Tom Heaton in the Cardiff goal with a tame free-kick from 20 yards in the third minute, the hosts went within inches of taking an early lead.
Mason, sent clean through on goal, saw his left-footed effort from 15 yards beat Burnley goalkeeper Lee Grant but graze the outside of the post on its way out of play.
In a lively opening to the encounter, Burnley quickly countered and Alex MacDonald saw his shot tipped wide by Heaton before Brian Easton rose highest to nod into the Bluebirds' stopper's arms once again.
But despite a promising start, the tie soon became a little flat, with each midfield cancelling the other out.
Conway pulled the trigger from distance without luck for Cardiff, while Jay Rodriguez headed Kieran Trippier's centre over the bar for the visitors, as neither side could find the breakthrough.
Andrew Taylor delivered a crisp diagonal ball into the box, Mason cushioned a header into the path of an unmarked Cowie and the Scotsman should have done better than wildly fire his shot high over the crossbar.
But the Bluebirds finally made the most of their chances six minutes before the break. Mason, dangerous throughout the first half, picked up possession on the edge of the Burnley penalty area, checked inside on to his left foot and produced a superb curling effort to beat Grant and hand Cardiff the lead.
After half-time neither side could find any momentum and, although Mason saw a powerful strike which found the back of the net ruled out for offside, the match began to turn stale.
It took until 70 minutes for Burnley to manage a noticeable attempt in the second period, but even then Mee could not direct his header from Ross Wallace's ball on target.
Marney's deflected free-kick went agonisingly close to levelling the game as the Clarets picked up in the match's final quarter, before Mark Hudson glanced a neat header marginally wide for the hosts with 15 minutes remaining.
Burnley ended the game on top but could not breach the Cardiff defence.
Cardiff Heaton, McNaughton, Hudson, Gerrard, Taylor, Whittingham, Cowie, Conway (Kiss 70), Gunnarsson, McPhail, Mason.
Subs Not Used: Marshall, Earnshaw, Keinan, Quinn, Blake, Ralls.
Booked: Gerrard, McNaughton.
Goals: Mason 40.
Burnley Grant, Trippier, Edgar, Mee, Easton, McCann (Wallace 46), Marney, Treacy (Austin 65), MacDonald (Bartley 46), Rodriguez, Hines.
Subs Not Used: Stewart, Amougou, Hewitt, Harvey.
Booked: Marney, MacDonald, Hines, Bartley.
Ref: Darren Deadman (Cambridgeshire).