Daizen Maeda's double, plus strikes from Callum McGregor, Adam Idah and Jota set up a meeting with Aberdeen on May 24.
Celtic are on course for a fifth domestic treble in eight seasons.
Brendan Rodgers' men lifted the League Cup in December and need just one more point from their remaining five games to secure a fourth consecutive league title.
Rodgers had been scathing of his players' attitude when they lost to bottom-of-the-table Saints in the league earlier this month.
There was never any chance of a repeat once McGregor was given too much space and time to fire in the opener on 34 minutes.
The floodgates then opened as McGregor teed up Maeda to curl in three minutes later.
Idah had a simple task to turn in Reo Hatate's cross to make it 3-0 before Maeda smashed in his 33rd goal of the season in first-half stoppage time.
There was only one more goal after the break as Jota volleyed in Alistair Johnston's cross just minutes after coming off the bench.