Steven Milne and Kenny Deuchar both scored twice, with a long-range Felipe Morais strike and a Collin Samuel volley sealing the win.
The result was never in doubt as St Johnstone took an early grip on the game at Gayfield with their main threat coming from defender Stuart McCaffrey, who twice hit the roof of the net with overhead kicks inside the first 15 minutes.
St Johnstone continued to press for an opener and Graham Gartland met a Liam Craig cross only to head into the side-netting.
It seemed only a matter of time before the Perth side took the lead and, after Milne forced a save out of Darren Hill with a shot on the turn, they went ahead on 24 minutes.
A slick three-man move, involving Danny Grainger, Craig and Milne culminated in the latter driving the ball low under Hill's body.
Moments later, Milne came close to doubling his tally when his audacious chip cracked the woodwork.
St Johnstone were now sensing a real chance to kill off the tie and they made it 2-0 on 27 minutes courtesy of former Hibernian midfielder Morais.
There appeared to be no immediate danger as Morais picked the ball up some 30 yards from goal but the Portuguese star tried his luck and hammered the ball beyond the reach of Hill.
The game was now turning into a decidedly one-sided affair and it was no longer a matter of if St Johnstone would score again, but when.
That question was answered on 42 minutes as Martin Hardie's low cross was met by Deuchar on the turn, with the striker hammering the ball under Hill's body.
And Saints underlined their superiority just before half-time as Milne lobbed into the top corner of the net from the edge of the box.
Derek McInnes' men were equally as commanding in the second period and nearly got their fifth goal on 52 minutes.
A finely-weighted Paul Sheerin pass was controlled on the edge of the box by Dave Mackay, who forced Hill into a decent stop.
Deuchar headed in St Johnstone's fifth on 67 minutes from Mackay's cross and the emphatic win was complete on 71 minutes as substitute Samuel volleyed beyond Hill.