James Fowler almost gave Kilmarnock the lead with just 17 seconds on the clock but shot wide after his first effort was blocked.
The visitors were then denied an early goal when Craig Bryson was well placed to head Ian Maxwell's effort off the goal-line.
St Mirren striker Mehmet came close in the 17th minute but his shot flew over the crossbar.
The home team went ahead when Hamill headed in a Hay corner, with Buddies goalkeeper Smith and his team-mates furious not to be awarded a free-kick for the midfielder's challenge.
St Mirren pushed hard for an equaliser before the break but Stewart Kean headed Mark Corcoran's cross wide and Mehmet hit another shot over bar before Alan Combe got down smartly to push away a strike from Andy Dorman.
Kilmarnock began the second half brightly and almost snatched a second goal after 55 minutes when Smith made a meal of Bryson's shot after the Saints goalkeeper had mis-hit a clearance.
Saints almost got back on level terms but Mehmet's header following Maxwell's throw-in went just wide of Combe's goal.
Invincibile almost doubled Kilmarnock's lead after 69 minutes but Smith produced a good block to deny the striker from 15 yards.
Allan Johnston then raced clear but he fired his cross behind with three Kilmarnock players waiting in the box.
St Mirren striker Jim Hamilton came close to scoring an own goal when he almost sliced a Bryson cross beyond Smith.
Invincibile should have tied things up for the home side seven minutes from time, but he drilled his shot low past Smith's right-hand post.
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