Kilmarnock's defence was under the cosh from early in the game and lost the opening goal when David Winters' cross broke to McCarthy who slid the ball home.
The SPL side's Colin Nish then fed Grant Murray, but his header was superbly saved by goalkeeper Bryn Halliwell.
McArthur doubled Accies' advantage after 37 minutes.
Lilley tried to clear a Mark Gilhaney cross only for the ball to unintentionally come off his hand.
Claims for a penalty were ignored by referee Kenny Clark but the ball broke kindly to McArthur who sent a right-footed shot low into the back of the net.
Kilmarnock tried in vain to reduce the deficit and Jim Jefferies' side's misery was compounded when David Fernandez received his marching orders after two quick yellow cards.
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