The hosts flew out of the blocks and went close with an early strike by Chris Chantler but there was only nine minutes on the clock when 18-year-old Muirhead smashed in a cracker from 25 yards for the opener.
Killie pushed on in spirited fashion as goalkeeper Dan Twardzik impressed with two saves from Josh Magennis to prevent his team from falling further behind.
Fraser Kerr was sent off with 15 minutes remaining to stifle the hopes of a comeback for the visitors.
And Kilmarnock made sure of the spoils from the resulting free-kick as Clingan rifled in hard and low to double the lead.
Motherwell boss Stuart McCall may bemoan the newly-installed controversial artifical surface at Rugby Park but he should have no complaints after an impressive show from Allan Johnston's men.