The Paisley side beat Hearts 3-2 at Hampden on March 17 to claim their first major trophy in 26 years. But as many as nine players named in the 16-man squad that day could be set to depart.
Irish right-back Van Zanten said: "It looks like there is going to be quite a big turnover at the club. I'm just glad I've got another year on my deal and that the manager has said that I'm part of his plans."
He added: "We had a cup-winning team but it looks like that is getting broken up now and there will be big changes over the summer. But we are looking forward to the new players coming in and the added competition they will bring."
Danny Lennon has already waved goodbye to keeper Craig Samson after he knocked back a new deal to move to Kilmarnock, while loan trio Conor Newton, Paul Dummett - both sent north by Newcastle - and Rio Ave striker Esmael Goncalves have all returned to their parent clubs.
Sam Parkin and Graham Carey have been released - as has Dougie Imrie, who was not stripped for the final - while Grant Adam and Lee Mair have been told to find new clubs.
And Paul McGowan - who struck a second-half penalty in between goals from Goncalves and Newton as Saints clinched victory - is also out of contract and on the verge of moving to England.
Saints were 15 points clear of bottom side Dundee when John Brown replaced Barry Smith in February. But they found themselves sucked towards a Clydesdale Bank Premier League relegation fight as the new Dens boss inspired a Dark Blues revival.
St Mirren survived in the end but Van Zanten, 31, says the club's alarming league slump has forced Lennon into a rethink.
He said: "We got a little bit of a fright at the end with Dundee closing the gap and the manager has looked at things and decided we need to improve. That's what he is doing now. We all want to improve our league position, we want to get in that top six."