Slater had made it 1-1 early in the second half at Rugby Park, but missed the chance to equalise again as he saw his penalty saved by Scott Brown in stoppage-time.
Adam Rooney had given the visitors the lead 40 minutes in when he connected with Donervon Daniels' flick-on following Niall McGinn's cross.
However, two minutes into the second half, Slater brought the hosts level as he curled an effort past Brown.
McGinn was again heavily involved as Cammy Smith restored Aberdeen's lead - the 19-year-old forward glancing a header from McGinn's delivery to register what proved to be the final goal of the game.
However, Slater missed the chance to salvage a point for Kilmarnock in the 91st minute after Brown brought down Miller inside the area.
The Aberdeen goalkeeper redeemed himself by saving Slater's weak effort as Derek McInnes' men equalled their league haul from last season and put pressure on leaders Celtic above them.
The defending champions could only muster a 1-1 draw with Inverness Caledonian Thistle on Saturday.
