The teams made the deal Tuesday, one day after Campbell had surgery on his broken collarbone that will sideline him at least six weeks.
The Bengals had been adamant that they wouldn't trade Palmer, who hasn't played since the end of last season. The franchise quarterback decided he wouldn't play another season with Cincinnati, even though his contract runs through the 2014 season.
But owner Mike Brown reversed course when the Raiders were willing to send a 2012 first-round pick and a conditional pick in the 2013 draft to the Bengals.
AP Sports Writer Joe Kay in Cincinnati contributed to this report.