"Josh is our starter and it's his job to lose," Wilks said.
Rosen will replace veteran Sam Bradford as the Cardinals' starter.
Rosen had his first action in a regular-season game on Sunday when he entered the contest after the Chicago Bears took a 16-14 lead with 4:31 left in the game. Wilks said Rosen would have entered the game earlier if the Bears had not held the ball for nearly seven minutes before Cody Parkey's field goal put Chicago ahead.
Wilks said Rosen handled the situation well and did a good job moving the ball. The coach said Rosen's mobility was also a factor in the decision.
Rosen, the 10th overall pick in this year's draft, led the Cardinals on a drive to the Bears' 45-yard line on his first series, but the drive ended when he threw an interception on a fourth-down play.
The game ended 16-14 in favor of Chicago.
Rosen finished 4-of-7 for 36 yards and the interception.
Bradford, who started the first three games, went 13-for-19 for 157 yards, two touchdowns and two interceptions, with a passer rating of 23.3.
On his last play, Bradford fumbled the ball away, setting up Chicago's go-ahead field goal.