Caley had the first serious effort on goal when Wilson curled a 20-yard free-kick just a foot wide of Roddy McKenzie's left-hand post.
Dennis Wyness tried to catch McKenzie out after the goalkeeper had mis-kicked his clearance, but Manu Dorado blocked his long-range effort.
Sam Morrow had Livi's first attempt on target after Dave Mackay had threaded the ball through, but Mark Brown was well positioned to clutch the ball into his chest.
A great run from David McNamee sent Richard Brittain clear on the left, but his low cross was cut out by Brown before it could reach Noel Whelan in front of goal.
Livingston's Sam Morrow, who had received treatment earlier after overstretching for the ball, had to go off after 26 minutes with Steve Hislop taking his place in attack.
Wilson broke the deadlock 11 minutes into the second half when he curled a free-kick over the wall and into the net.
Livi tried to hit back but failed to seriously trouble Brown in the Caley goal.
Whelan and Colin Healy both had efforts just wide but Brown did not have to do anything out of the ordinary. He did well to clutch a late free-kick from Wes Hoolahan under his crossbar, but that was as close as Livi came.