St Johnstone dominated for long periods and pushed Celtic all the way as they belied their relatively lowly league position.
Celtic's opener came after St Johnstone had made the first meaningful chance of the match. In the 11th minute, Artur Boruc did very well to deny Saints' top scorer Jason Scotland after he had been put through by Peter McDonald.
Two minutes later Celtic striker Jan Vennegoor of Hesselink was brought down by Saints keeper Kevin Cuthbert and the big Dutch striker scored the resulting penalty even though Cuthbert got a hand to it.
St Johnstone deservedly equalised in the 19th minute though when Martin Hardie, after latching on to a McDonald pass, lobbed the ball brilliantly over Boruc from an acute angle and Stephen McManus headed into his own goal attempting to clear.
After the break Celtic's midfield dynamo Paul Hartley raced towards the edge of the box and fired goalwards. Cuthbert saved his effort, but from the resulting 54th minute Shunsuke Nakamura corner, Hesselink powered home a close-range header.
St Johnstone bounced back though and dominated the bulk of the second half, with their more illustrious opponents only threatening on the break.
In the 64th minute Saints went close to grabbing an equaliser after a great three-man move involving Goran Stanic, Paul Sheerin and Scotland. Ten minutes later Scotland beat Boruc only for McManus to clear the ball off the line.
On the break Celtic looked dangerous at times and Aiden McGeady, threatening throughout, went very close in the 77th minute.
In the closing stages St Johnstone threw everything at the champions-elect but Gordon Strachan's men held on as they ground out another of the type of victories which may well land them a league and cup double at the end of the season.