It was a pretty poor contest throughout but the Bairns almost put the lead 45 seconds from the kick-off.
Anthony Stokes fired in a low right-foot shot from the edge of the box that forced Jamie Langfield to dive full stretch and turned round the post.
It was the home side again who came close with nine minutes played but Darren Barr's header smashed off the crossbar and bounced to safety.
That was as much goalmouth action as a bored crowd witnessed in the first half but ten minutes after the re-start Aberdeen took the lead with a quality goal.
Barry Nicholson's corner was headed out to Gary Dempsey 24 yards out and the midfielder sent a screaming right-foot shot low inside Scott Higgins' left corner.
Stokes was unlucky not to level things when his angled drive in the 73rd minute rattled off the junction of crossbar and upright but, with nine minutes remaining, the Dons made sure with a second.
Stephen O'Donnell and Barr barged into each other as the last line of defence and visiting sub, Dyron Daal, was left free to slot past Higgins.