Gary Harkins scored the only goal in the 26th minute of the McDiarmid Park contest to send 2,500 visiting fans back to Dundee in party mood.
Dark Blues manager Paul Hartley started with a skilful midfield of Simon Ferry, Paul McGowan, Gary Harkins and Jim McAlister – with Philp Roberts on the right flank – and his side dominated possession thanks to some impressive passing and movement.
They were also solid at the back with central defenders James McPake and Konrad helping ensure that goalkeeper Kyle Letheren did not have a save of note to make.
St Johnstone were struck by a double injury blow as manager Tommy Wright revealed just before kick-off that both Tam Scobbie and Steven MacLean might need knee operations. Frazer Wright, Michael O’Halloran and Brian Graham all came into the team, with Lee Croft dropping to the bench.
Dundee were also forced into a change, Ferry replacing Peter MacDonald after the former Saints striker was ruled out with a knee knock.
On-loan Dundee United striker Graham sent a diving header wide from an early corner, but Dundee soon took control.
Ferry blasted high and wide after a poor clearance from Frazer Wright before the breakthrough came in the 28th minute through a set-piece.
McAlister headed a Roberts corner back into the goalmouth and Harkins hooked the loose ball into the far corner.
Harkins then fired a first-time shot over from 18 yards as Dundee tried to build on their lead and they comfortably survived a brief spell of home pressure before McAlister saw a shot deflected wide after good wing play from Roberts.
The visitors remained on top after the restart and good skill from Harkins set up Greg Stewart – but the lively striker’s shot was blocked.
Referee Willie Collum pulled play back for a late tackle on Harkins and the midfielder’s free-kick was pushed away by Alan Mannus.
St Johnstone soon lost Scott Brown to injury, but the introduction of Simon Lappin failed to turn the tide and Paul McGowan had a shot saved from a tight angle.
The Championship winners continued to look comfortable with the home side only managing a few long-range efforts which failed to trouble Letheren, and some well-defended crosses.
Stewart and Harkins relieved the pressure at times with some enterprising runs and substitute Luka Tankulic headed on to the roof of the net as Dundee saw out the closing stages in the St Johnstone half.