Prescott fractured his right thumb when his hand struck the helmet of Tampa Bay pass rusher Shaquille Barrett with about six minutes remaining in the game.
Cowboys team owner Jerry Jones said the injury would keep Prescott sidelined for "several weeks", while ESPN has reported he will miss "six to eight weeks as he goes through rehabilitation". That length would see him out until November 13 against the Green Bay Packers.
Prescott missed most of the 2020 season with a fractured right ankle, and the Cowboys went 4-7 in his absence with three different quarterbacks making at least one start, with veteran Andy Dalton getting the bulk of playing time.
Cooper Rush, who stepped in after Prescott's injury on Sunday, is expected to make his second career start when the Cowboys host the defending AFC Champion Cincinnati Bengals next Sunday.
Cowboys head coach Mike McCarthy said: "The nice thing about Cooper is Cooper's been in our system, knows our system inside and out.
"We won't be in that position of trying not to do too much or vice versa. Cooper gives us the ability to keep playing [our offense]."