The away side's dominance continued in the opening gambit and they should have gone in front when Charlie Mulgrew rose highest in the box, only to send his header over the crossbar.
The attacks continued to come and a lively Griffiths almost made use of a well-placed free kick, but his effort struck the crossbar, much to the delight of stranded Aberdeen stopper Ward.
Aberdeen failed to produce much in front of goal, although when they did Andrew Considine could not capitalise as his close-range header somehow made its way over the bar.
The visitors finally got the goal their play deserved just before the break, when Griffiths stepped up to slot home a penalty after Considine had brought the forward down inside the Aberdeen area.
The hosts sought to press on from the leveller and they almost found a goal to put them ahead through Kenny McLean, but his header from inside the box flashed over the bar.
The Hoops almost made the one-man advantage count in the closing stages, although James Forrest saw both of his attempts kept out by a valiant Ward.