The tie was evenly poised following a 1-1 draw in Lucerne last week, but St Johnstone were in the driving seat after Stevie May’s penalty put the home team in front midway through the first half.
May coolly converted his spot-kick on 23 minutes after midfielder Craig Miller was hauled down in the box by Xavier Hochstrasser.
Luzern had a penalty shout of their own turned down by referee Jonathan Lardot when Dario Lezcano threw himself to the ground five minutes into the second half.
But the Swiss side forced an equaliser in the 58th minute when substitute Marco Schneuwly, the scorer of the equaliser last week, bundled the ball home.
Schneuwly and May both had chances to win the game in 90 minutes but neither were able to convert at 1-1, forcing the tie into extra time.
Both sides had chances to win it in extra-time with the best of them falling to St Johnstone substitute Liam Caddis who hit the post in the 118th minute.
After both teams scored their opening penalties in the shoot-out, Mannus brilliantly clawed away Schneuwly’s effort to give St Johnstone the advantage.
And the home side didn’t look back from there as Steven MacLean, Caddis and Dave Mackay all converted before Tam Scobbie scored the all-important fifth penalty to send St Johnstone through.