The teenage midfielder netted after just four minutes with a precise drive which went in off the post, and doubled his team’s advantage 10 minutes after the restart with a fabulous finish from Richard Offiong’s cross.
Kilmarnock, who were forced to substitute defender David Lilley and forward Danny Invincibile due to injury inside the first 15 minutes, pulled a goal back with an injury-time penalty from Jamie Hamill.
Hamilton welcomed back Offiong from a knee injury, along with Scotland B squad defender Brian Easton, Joel Thomas and James Gibson making way by dropping to the bench.
Mark Corcoran was also among the substitutes after nearly six months out through injury.
Kilmarnock’s team was unchanged from that which started against Dundee United two weeks earlier.
Hamilton took the lead on four minutes after a lively opening which had seen Offiong and Lilley poleaxed by a clash of heads inside the first 30 seconds.
Simon Mensing passed the ball out to McCarthy and the teenager drilled home a precise diagonal shot which went in off the right-hand post.
Kilmarnock were forced into two early substitutions as Lilley had to be replaced in the centre of defence by Frazer Wright, and Invincibile suddenly pulled up and limped off after 15 minutes.
By then Hamilton had come close again via a looping Mensing header which Alan Combe clutched above his head despite Offiong’s efforts to nod it home on the line.
Kilmarnock struggled to threaten Tomas Cerny’s goal after that early disruption to their team, but finally enjoyed more pressure in the 15 minutes before the interval.
David Fernandez released Mehdi Taouil with a pass from midfield and the Moroccan worked his way down the left flank and tried a shot which the home goalkeeper easily smothered at his left-hand post before Jamie Hamill was able to slide in and connect.
Fernandez then tried a turn and shot in a busy box but the effort never really troubled Cerny, while Paul McGowan immediately came close at the other end with a shot which sneaked marginally wide of the left-hand post.
Their best effort fell to Hamill from a Willie Gibson cross from the left byline, but Cerny claimed his volley from the centre of the six-yard box.
Hamilton made a change just minutes into the second half when McGowan was replaced by Derek Lyle, but it was Kilmarnock who had the first sight of goal when Garry Hay rolled the ball across the face of goal and it was played back out to Tim Clancy for a long-range looping effort which Cerny caught.
Kilmarnock looked promising in the early stages of the half but it was the hosts who doubled their advantage on 55 minutes.
Offiong found McCarthy with a cross which the teenager expertly lashed home from just inside the area.
Kilmarnock responded soon afterwards by withdrawing defender Clancy to bring on Connor Sammon to boost the forward line as they attempted to salvage something from the game.
Bryson sent a hopeful drive well over the crossbar in the final few minutes but Kilmarnock were handed an injury-time lifeline when a penalty was awarded against Easton.
A looping ball dropping from the air hit the defender’s arm as he attempted to head clear and Hamill sent Cerny the wrong way from the spot.