Since taking over at St Mirren Park in 2010, Danny Lennon has failed to manage a point or even a goal against the Clydesdale Bank Premier League champions.
Earlier in the season Celtic strolled to a 5-0 away victory against the Paisley outfit and van Zanten insists the pressure is on to avoid a similar result when the two sides meet on Saturday.
"The pressure is always on no matter who the opposition are, you are always under that pressure to win the game," said the 30-year-old defender, who began his career as a youth player at Celtic Park.
"These are the games you look forward to, playing in front of big crowds and against the best players you possibly can play against.
"We know we will need a bit of luck and to be at our best - hopefully that will be enough to get the three points.
"Our aim is to go there and try win the game, albeit by stopping Celtic first and foremost and then try to impose ourselves on the game."
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