Gary Liggett's treble gave Whites boss John Cunningham his first win in charge at New Grosvenor.
The striker fired in a 20-yard opener before rounding keeper Declan Brown and slotting in after 26 minutes.
Paul Heatley burst through and scored for the visitors but Liggett sealed the points and his treble with another cool finish with 10 minutes left.
Carrick ended with nine men with Andy Smith dismissed for dissent and Anto Lagan red-carded after a poor tackle.
Distillery are now level on points with Glenavon but they remain bottom on goal difference.