He was injured when jumping for a high ball with Peru's Jefferson Farfan and the 33-year-old Kvist had to be carried off the pitch on a stretcher and substituted.
"William Kvist has suffered a rib injury and is travelling home to Denmark. Everyone around the team wishes him a good recovery," the Danish FA said on Twitter.
The victory put Denmark in second place in Group C on three points with France top on goals scored. The Danes take on Australia in their next match on Thursday.
(Reporting by Philip O'Connor, editing by Ed Osmond)
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