PA Sport The only goal came in the 25th minute when Stephen Hughes left a back-pass short and Eljero Elia rounded goalkeeper Jason Brown and slotted the ball into the empty net.
Naysmith had to be at his best to prevent Theodor Gebre Selassie converting Elia's low cross as they remained only a goal behind at the interval.
Another defensive lapse almost let Aaron Hunt in on the hour, but his shot was deflected wide.
Aberdeen's best chance came when Scott Vernon met Ryan Fraser's cross and the ball broke to Mitchel Megginson, but his effort was blocked.
