The Danish international signed a five-year deal at Palace after joining from Lyon in a deal worth up to £17million.
Andersen had previously been a target for Tottenham after impressing in the Premier League while on loan at Fulham last season, but says talks with new Palace boss Vieira convinced him to join the Eagles.
Palace are keen to shake up their playing style following the departure of Roy Hodgson and their summer business so far has pointed to a far more progressive outlook.
Andersen is one of five new additions so far alongside Conor Gallagher, Marc Guehi, Michael Olise and Remi Matthews.
“I had several talks with him [Vieira],” said Andersen.
"What has been said I will keep for myself, but obviously he gave me a good feeling and he told me how he wanted to play. That matches my football view and the way I want to play football, and that’s fantastic.
“I want to play football from the back, and he wants that as well. So I think it fits really well. It’s quite exciting to start up something new with new players and a new style, so we have to learn – everyone – and hopefully I can be a big part of us moving forwards faster.
“I want to show the same [things] that I showed last season. My ball-playing skills, some long balls played out from the back, my defensive skills, leading the back line and leading the team. So it’s something I can bring to this team hopefully.”
Speaking to the Palace website, Andersen added: “I think it’s a really interesting project going on here with the new manager. Many new players coming in, a new style of play. Obviously I know Palace from playing in the Premier League last season, so I know it’s a big club – a London club.
“I’m just really looking forward to this new project and playing under a manager that wants to play football in a style that suits me well.”