The Blues raced into a 17-0 lead by the interval in muddy conditions at the Arms Park.
The visitors were made to pay a heavy price for their indiscipline, with two sin-binnings during the half for Dafydd Jones and Ceiron Thomas.
Nicky Robinson put in a clever grubber kick for Czekaj to win the race for his first try in the 29th minute.
Five minutes later Robinson was again involved in the second try linking with his brother, Jamie, to mount a thrilling attack from their own 22 and the move was finished off by teenage wing sensation Tom James.
The Blues’ dominance was underlined when Scott Morgan finished off a textbook catch-and-drive from a close range line-out that Blair converted.
Llanelli eventually got some reward when the home front row all infringed at a scrum and the referee awarded a penalty try that Ceiron Thomas converted. The Blues finished strongly when Robinson’s long pass put Czekaj over for the final try.
Cardiff Blues scorers: Tries: Czekaj 2, James, Morgan. Cons: Blair 3. Pens: Blair.
CARDIFF BLUES: Blair, James, J. Robinson, Stcherbina, Czekaj, N. Robinson, Phillips, Jenkins, Thomas, G. Powell, Jones, Davies, Morgan, Shellard, Rush.
Replacements: Macleod for J. Robinson (64), Yapp for Jenkins (71), Goodfield for Thomas (71), Goode for Jones (69), White for Shellard (57).
Llanelli Scarlets scores: Tries: Penalty. Cons: C. Thomas. Pens: C. Thomas.
LLANELLI SCARLETS: G. Evans, James, King, Watkins, M. Jones, C. Thomas, Peel, I. Thomas, Rees, Manu, Cooper, MacLeod, D. Jones, Bater, N. Thomas.
Replacements: Daniel for James (53), G. Evans for Watkins (41), G. Thomas for Bater (41).
Referee: Hugh Watkins (Wales).