The former Arsenal captain emerged as Palace’s preferred candidate after a lengthy process since Hodgson announced he was leaving the club on 18 May.
The 45-year-old Frenchman has signed a three-year contract with the Eagles, who finished 14th last season.
“I am really excited to have this opportunity to return to the Premier League,” said Vieira.
What kind of manager are Palace getting in Vieira?
“It is a project that is really appealing to me, having spoken a lot with the chairman [Steve Parish] and sporting director [Dougie Freedman] about their ambition and plans for the whole club, including the academy.
“The club has fantastic foundations in place after many years in the Premier League, and I hope we can make further improvements and continue to drive the club forward.”
Vieira was sacked by French club Nice in December.
Previously he had spent two and a half years in charge of Major League Soccer side New York City after beginning his coaching career in the academy at Manchester City, where he ended his stellar playing career in 2011.
He secured two play-off appearances at New York City and achieved seventh- and fifth-placed finishes during his two full seasons at Nice.
Former Borussia Dortmund boss Lucien Favre held talks over the role but pulled out of the running to succeed the 73-year-old Hodgson, whose contract ended in the summer after four years in charge.
Parish said he was “delighted” with the appointment following Vieira “successfully learning his trade at the City Group, and experiencing positive managerial spells at New York City and then Nice, who he led to the Europa League”.
He added: “We have a lot to do to prepare in the coming weeks and I look forward to working closely with Patrick and Dougie in bringing in some new faces to help us all to a successful season.”
France midfielder Vieira claimed seven major honours in nine years as a player at Arsenal between 1996 and 2005.
A World Cup and European Championship winner with France, he also helped Inter Milan to three Serie A titles.
Vieira joined Manchester City in January 2010 and helped the club lift the FA Cup the following year before announcing his retirement from playing.