Ilian Kiriakov had the Dons first on target shot, shooting well from 20 yards.
Phillip Scott had the visitors first goal attempt in 16 minutes but he failed to hit the target with his effort.
Aberdeen man Paul Kane controlled before driving a low shot past Leighton's left-hand post.
Miguel Simao finished coolly seven minutes from the break after Paul Kane had carved open the Aberdeen defence.
An Eoin Jess drive was as close as the Dons came to scoring in a dreadful second half, which saw St Johnstone hold on to their lead.