Marcus Cooper, acquired off waivers from Kansas City on Sept. 2, had two interceptions, returning the second 60 yards for a score. Winston also fumbled the ball away once.
Winston, who had four touchdown passes against Atlanta in the opener and was NFC player of the week, threw 51 passes, the most in his two NFL seasons, completing 27 for 243 yards.
Palmer completed 18 of 31 passes with no interceptions before sitting out most of the fourth quarter.
The Cardinals (1-1), coming off a season-opening home loss to New England, scored two touchdowns in a 1:16 span in the final two minutes of the first half for a 24-0 lead into the break.
Winston had an especially rough first 30 minutes, completing 11 of 20 for 77 yards with two interceptions and a fumble.
The four turnovers led to 17 Arizona points.
The Buccaneers got the ball back and were driving again when defensive lineman Josh Mauro tipped Winston's pass. Cooper, who replaced rookie Brandon Williams early in the game, grabbed it and outran defenders down the sideline to make it 33-7.
The Buccaneers were on their best drive of the first half when running back Charles Sims III brushed against Winston and the ball popped loose. Tony Jefferson recovered for Arizona at the Cardinals 29.
QUICK-STRIKE OFFENSE: After the fumble recovery, Palmer's 24-yard pass to Fitzgerald eventually put the ball at the Tampa Bay 1. On third down, Palmer hit Michael Floyd in the back of the end zone and it was 17-0 with 1:36 to go in the half.
A quick three-and-out by the Buccaneers gave the Cardinals the ball at their 26, plenty of time for a high-octane offense finally getting in gear.
Jaron Brown got behind the defense and gathered in a perfect pass from Palmer for a 49-yard touchdown with 20 seconds still remaining until halftime.
WINSTON PICKED: Peterson outmaneuvered Mike Evans on Winston's deep pass to the end zone and intercepted .
Cooper returned an interception 23 yards to the Tampa Bay 19, leading to an Arizona field goal.
The third pick was Cooper again, for a touchdown. Winston's fourth interception came on the game's last play.
INJURIES: Injuries hit the Buccaneers hard.