GLENDALE, Ariz. -- Adrian Peterson did more in his first two quarters Sunday with the Cardinals than he could in his first four games combined with the Saints, but it almost didn't matter as Arizona had to hold off a furious rally before vanquishing the Tampa Bay Buccaneers, 38-33.

The Cardinals scored points on each of their first five possessions of the game in beating the Buccaneers for the second straight year at home. For Tampa Bay, this one was worse than last season's 40-7 loss, as quarterback Jameis Winston exited the game with an injured right shoulder in the second quarter and did not return.

Ryan Fitzpatrick replaced Winston and threw three touchdown passes.

Winston was injured on the Buccaneers' first series of the game when he scrambled outside of the pocket looking to throw and was hit by pass rusher Chandler Jones. X-rays were negative, but head coach Dirk Koetter said the team likely won't know the full extent of the injury until Monday.

"He's not the first or the last quarterback that's going to have an injury," Koetter said. "Obviously, there was a big injury in Green Bay as well today. Injuries are an unfortunate part of the game. Jameis is as tough as they come. This isn't about toughness. This is about he got a shoulder injury and couldn't throw the football."

"I'll be back soon," said Winston, who was 5 of 10 for 61 yards before leaving.

The Cardinals led 24-0 when Winston exited.

Peterson scored on a 27-yard touchdown run, Carson Palmer threw touchdown passes to tight end Troy Niklas and wide receiver Larry Fitzgerald, and Phil Dawson kicked a 28-yard field goal.

Palmer added a 17-yard touchdown to wide receiver John Brown to make it 31-0 four minutes into the third quarter.

"There's no lead big enough in the National Football League," said Cardinals head coach Bruce Arians, whose team improved to 3-3. "We shouldn't have to be biting our nails off to get an onside kick. You have to keep your foot on the gas."

And indeed, the Buccaneers mounted a major comeback.

Fitzpatrick threw touchdowns to wide receiver DeSean Jackson and tight end Cameron Brate on consecutive possessions and then Tampa Bay linebacker Lavonte David forced a fumble by Fitzgerald and returned it 21 yards for a touchdown.

After getting back in the game, Brown caught a 31-yard pass on a third down from the Arizona 37 and the Cardinals consumed too much of the clock.

Though the Bucs forced the Cardinals to punt, Andy Lee's kick and Budda Baker's catch pinned Tampa Bay at its own 1-yard line. Moments later, Fitzpatrick tried forcing a short pass to wide receiver Mike Evans and it was intercepted by safety Tramon Williams.

Peterson turned the turnover into a touchdown with a 1-yard scoring plunge and it was 38-20 Cardinals.

Fitzpatrick mounted yet another drive, capped by Doug Martin's 11-yard touchdown run. The two-point conversion attempt failed.

Fitzpatrick later would find Evans for a 37-yard touchdown and with the extra point, the Bucs pulled to within five points with 2:03 left.

"No moral victories in the NFL," Fitzpatrick said. "We just didn't get it done. I made a couple big mistakes. ... There's a couple throws I'd like to have back that really cost us, but I was proud of the way the guys fought."

Palmer completed his first 14 passes and he was 18 of 22 for 283 yards, three touchdowns and the one interception

Peterson ran 26 times for 134 yards and two touchdowns in his Cardinals' debut. In four games with the Saints, he gained 85 total yards on 27 carries. Peterson, acquired in a trade earlier in the week for a conditional sixth-round pick in 2018, ran for more than that by halftime, rushing for 85 yards on just 11 carries.

"You could feel it that first drive," Palmer said when asked what type of impact Peterson made on the team. "You could feel it with the fans in the stadium. You could see it on the defense's face, body language-wise, it was a great lift for us. The defense was feeling it. Everybody was excited and it was great to see."

Peterson gained 54 yards on just four carries during the first drive of the game.

"I feel good. This is the opportunity I have been waiting for," he said, adding that opening drive paved the way for a big day. "Personally, it gave me the opportunity to get my lungs open. You can go through practice, running and running 40 yards down the field, but it's nothing like avoiding defenders and bursting, trying to get to the end zone every play. It got me warmed up."

Fitzgerald caught 10 passes for 138 yards, including eight receptions for 128 yards in the first half.

Fitzpatrick completed 22 of 32 passes for 290 yards, three touchdowns and two interceptions.

NOTES: Cardinals RB Adrian Peterson's two rushing touchdowns tie him with Hall of Famer Barry Sanders (99) for ninth on the NFL's all-time list. ... In his last visit to University of Phoenix Stadium, Buccaneers rookie TE O.J. Howard caught five passes for 208 yards and two touchdowns for Alabama in the national championship game against Clemson. ... With his 29-yard reception in the first quarter, WR Larry Fitzgerald surpassed Steve Smith Sr. (14,731) for seventh place on the league's all-time receiving yards list and also reached 14,735 career receiving yards, leaving him behind only Jerry Rice (19,247 with the 49ers) for the most receiving yards by any receiver with the same team. ... CB Patrick Peterson aggravated a sore quadriceps muscle in the fourth quarter and did not return.