Kvist left the pitch in the 34th minute at the Mordovia Arena in Saransk.
Ajax's Lasse Schone was brought on to replace Kvist.
The Danes' No. 7 collided with Peru forward Jefferson Farfan. Kvist appeared to catch a knee in the ribs from the 33-year-old Lokomotiv Moscow forward.
Losing Kvist is a blow for Denmark coach Age Hareide. The veteran defensive midfielder only played sparingly for Copenhagen this season, but his experience and tenacity in the middle helps shield the defence.
His ability to anchor midfield also affords Tottenham Hotspur playmaker Christian Eriksen greater freedom to get forward and probe for chances.
Fortunately for Hareide, Schone is a more than worthy replacement for former Wigan Athletic, Fulham and VfB Stuttgart man Kvist.
Schone made 35 appearances across all competitions for Ajax and found the net 11 times. He also chipped in with three assists. Schone may offer more to Denmark's play going forward, even if Hareide is likely to miss Kvist's discipline and savvy.
The Danes will hope an important squad man isn't lost for the tournament.