Cardinals coach Steve Wilks named Rosen, Arizona's first-round draft pick in April, the starting quarterback against the Seattle Seahawks during his Monday afternoon news conference.
"Josh plays with a lot of confidence," Wilks said. "I think he gives us an opportunity to be successful. When you look at the situation when he went into the game, I didn't have a problem putting him in at that time because, again, I know he's very confident in what he's doing."
Wilks made the decision to start Rosen on Monday morning and informed the team during a meeting. The team's reaction was "fine."
"Josh was excited about the opportunity and I'm excited for him," Wilks said.
Wilks said he didn't get any pushback on the move from the front office.
"As I talked before [with] Sam, I said Sam was our starter and it was his job to lose, and I'm saying the exact same thing about Josh Rosen," Wilks said. "Josh is our starter and it's his job to lose."
Rosen replaced Bradford during Sunday's 16-14 loss to the Chicago Bears with 4 minutes, 31 seconds left in the game and played the final two drives. He attempted to march the offense into range for a game-winning field goal but was intercepted to end his first drive, then was sacked to the end the game.
Wilks said he decided to play Rosen on Sunday after Bradford lost a fumble with 11 minutes, 33 seconds left in the game, but the Bears' next drive took 6:52, leaving Rosen with just over 4 minutes.
"It looked like Josh was thrown in at the end but he wasn't," Wilks said. "So, he can handle those types of situations and I think he handled it well. When he went in, commanded the huddle, did a great job of moving the ball."
Wilks praised parts of Rosen's performance Sunday, pointing out his accuracy and ability to get first downs.
"He's going to have some growing pains," Wilks said. "That happens as a rookie and you're going to see some great things from him as well."
Wilks said Rosen's toughest challenge as the starting quarterback will be understanding the protections at the NFL level.
"You see so many different looks at this level," Wilks said. "And being able to identify those looks on the run is going to be important. That's why I think the communication up front with the offensive line is going to be key."
Rosen, 21, was picked 10th overall by the Cardinals.
There now will be four first-round rookie quarterbacks starting in Week 4 (Baker Mayfield, Sam Darnold, Rosen and Josh Allen). The last time four first-round rookie QBs all started in the same week was Week 16 of 2012, when it was Andrew Luck, Robert Griffin III, Ryan Tannehill and Brandon Weeden.
ESPN Stats & Information contributed to this report.