The striker scored an emphatic header four minutes before half-time to maintain the Championship leaders' 100 per cent record for the season, while top-flight Kilmarnock were eliminated at the second-round stage for a second successive year.
Antoine-Curier's starting berth alongside Tony Andreu was one of two changes from Accies' third straight Championship win three days earlier, with the pair respectively starting in place of Jon McShane and player-manager Alex Neil, both of whom were on the bench.
Kilmarnock manager Allan Johnston made three changes to the team which had lost to Hibs three days earlier, with William Gros, Kyle Jacobs and Rory McKeown starting in place of James Fowler, Barry Nicholson and Paul Heffernan, who were all substitutes.
Hamilton took the initiative in the early stages, busily finding their way forward in the opening spell and earning a series of corners, including one after Darian MacKinnon burst with pace into the box on seven minutes.
Rabiu Ibrahim had a half-chance for Kilmarnock in the first quarter with a diagonal effort from the left corner of the six-yard box, but the ball ran across goal and out to safety.
Accies looked the likelier side to open the scoring in the first half and came close on 18 minutes when Ali Crawford sent in a free kick from the right wing and Michael Devlin rose well in a busy penalty box to head just over the bar.
The hosts carved a half-chance on 30 minutes after Ibrahim skilfully retained possession and escaped his markers at the left corner flag. He passed to Rory McKeown and the ball was then cleared out to Sean Clohessy at the right corner of the box, but his shot failed to trouble Cuthbert.
Instead, Hamilton took a deserved lead four minutes from the break in simple but emphatic style - Crawford delivered a corner from the right and Antoine-Curier rose highest in the box to bullet a header past Samson and in at the near post.
Kilmarnock had their best chance of an equaliser five minutes after the restart when Ibrahim unleashed an impressive shot from outside the box, which Cuthbert had to claw away round the right-hand post.
Gros also had an early chance and Clohessy shot straight at the goalkeeper as the hosts had their best spell of pressure shortly after the restart.
Accies also had second-half efforts thanks to Antoine-Curier and James Keatings, who were replaced 20 minutes from time by McShane and Andy Ryan.
Clohessy then sent in a cross 12 minutes from time which player-manager Neil, on as a substitute, blocked with a header; and from a second corner, Gros headed narrowly wide.
Kilmarnock had some late efforts on goal but were unable to find a late equaliser to force extra time, while Andy Ryan hit the side netting for Hamilton in stoppage time.