Since taking over at St Mirren Park in 2010, Danny Lennon has failed to manage a point or even a goal against the Scottish 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."
The Buddies will make the trip on the back of four games without defeat.
They will also be encouraged by Celtic's poor home form with the Hoops having failed to record a home win in the league since early October.
St Mirren endured a run of six successive defeats between October and the middle of November, however after tightening up in defence they have moved six points clear of Dundee at the foot of the table.
"It may have been circumstances, but all we can do is take hope from the results that other teams have had there this season and aim to replicate that," van Zanten added.
"We have passed up a lot of opportunities recently, it's the other way round from earlier on in the season where we were conceding a lot of goals but were taking our chances too.
"The next three or four games present a great chance for us to put some points on the board and move up the table, there was never a worry during our poor form and everything was positive."
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