Second-half goals from Niall Quinn and Mark Stafford booked Linfield's place in the County Antrim Shield final against Ballymena United on 12 January.
Quinn showed composure to sidefoot a low shot into the bottom corner after Paul Smyth picked him out with a cross from the left in the 54th minute.
Defender Stafford added the second by heading in powerfully from six yards out from a corner 20 minutes later.
The game had been due to be played at Taylor's Avenue two weeks ago.
The match was called off on that occasion because of the impact of the weather on the pitch and the tie was subsequently switched to Windsor Park to avoid the possibility of a second postponement.
Andrew Waterworth directed a right-foot shot wide of the post early on and then saw his penalty saved low to his left by Brian Neeson after the Carrick goalkeeper had brought down Smyth.
Neeson parried and then gathered a Kirk Millar effort, while Smyth fired into the sidenetting after running through on goal.
After half-time, Kyle Cherry was off-target with a left-foot strike and soon after Quinn broke the deadlock with his first goal of the season.
Former Ballinamallard player Stafford made victory secure after being left unmarked in the area.