When Saints equalised early in the second half through Stephen McGinn, the grandson of former Celtic chairman Jack McGinn, it looked likely that the visitors could create an upset.
However, after Shunsuke Nakamura had been brought down for the penalty, there was only one outcome as Celtic added further goals at will.
The visitors created an early chance when Alan Reid headed Simon Lappin's cross back across goal, but Steven Pressley stepped in to clear.
Saints keeper Chris Smith blocked Craig Beattie's snap shot with his legs, then did even better to turn away Thomas Gravesen's volley from a Nakamura corner. However, Beattie should have scored after the keeper's parry, but he shot high over.
Celtic took the lead when Nakamura's corner found Vennegoor of Hesselink unmarked on the six-yard line and the striker headed powerfully home.
The home team were shocked early in the second half when Saints equalised. Mark Corcoran touched on a cross from Lappin on the left to McGinn, who steadied himself before shooting into the top corner from 18 yards.
The visitors took encouragement from this equaliser and Celtic went through an uncertain spell before Ian Maxwell clumsily challenged Nakamura from behind and Vennegoor of Hesselink stroked the penalty home.
McGeady made it three when he drilled home a Beattie cut back before being substituted by Derek Riordan.
Vennegoor of Hesselink claimed his hat-trick when Nakamura's clever reverse ball found Beattie and from his cut back with his back to goal, the striker swivelled before shooting low into the corner of the net.
There was still time for Miller to notch a fifth when Riordan's blocked effort fell to him and he bundled the ball home from close range.