Behrami's shot took a deflection off Sunderland striker Kenwyne Jones to beat stranded keeper Marton Fulop.
Kenwyne Jones had headed Stoke into the lead after six minutes when he converted Jermaine Pennant's corner.
The hosts should have been cruising at half-time, but Liam Miller and Kenwyne Jones went close as Sunderland rallied.
The Reds should have gone behind but Kenwyne Jones had his shot saved when one-on-one with keeper Jose Reina.
The visitors looked lively as Kenwyne Jones hit the bar on three minutes before injury forced his early exit.
Crouch headed his second before Kenwyne Jones' strike kept Stoke in the game, but further goals eluded both sides.
Substitute Kenwyne Jones made it 3-3 but Santa Cruz's second won it for City before Michael Turner was sent off.
Kenwyne Jones headed in late on for the Black Cats, who stayed up after rivals Newcastle lost 1-0 at Aston Villa.
He found himself behind Kenwyne Jones, Djibril Cisse, Daryl Murphy and David Healy in the Premier League side's pecking order last season.
Kenwyne Jones pounced on an error from Blackburn goalkeeper Paul Robinson to level before Djibril Cisse converted a Steed Malbranque pass for the winner.
But Danny Collins and Anthony Stokes failed to find the target with flashing first-half headers and Kenwyne Jones could not turn in a left-wing cross.
However, the Black Cats have an abundance of talent in their ranks, not least of which is their front three of Kenwyne Jones, Bent and Campbell.
Ferguson's side put in a poor performance, with keeper Ben Foster at fault for the second-half header from Kenwyne Jones, which put the visitors 2-1 ahead.
But, Kenwyne Jones aside, they were unable to find a way through a makeshift City back-line that was missing Micah Richards and Nedum Onuoha through injury.
He hit out at strikers Djibril Cisse and Kenwyne Jones, claiming they "did not have the spark up front" after the 3-0 defeat at Everton on Saturday.
And Spurs continue to be linked with West Ham striker Craig Bellamy and Sunderland forward Kenwyne Jones as they look to strengthen to ensure their Premier League survival.
Bent had a snap-shot easily saved by Petr Cech, while Daryl Murphy was bravely denied by the keeper from eight yards out after good work by Kenwyne Jones.
But there were several chances for the home side to make the game safe, substitute Kenwyne Jones completely missing his kick in front of goal and Zenden seeing two efforts blocked.
Striker Kenwyne Jones was forced off at half-time for Sunderland after injuring his wrist when painfully blocking a Gilberto shot and his replacement O'Donovan almost put his side level when play resumed.
Grant Leadbitter had Sunderland's first shot in anger as he struck a 22-yard effort wide before Kenwyne Jones just failed to get a crucial headed touch to divert cross past keeper Petr Cech.
And the addition of an extra striker turned the match in their favour, with Kenwyne Jones heading an equaliser before a late own goal from James Perch gave them their first away points of the season.
Saturday's evidence would suggest manager Mick McCarthy could expect to more than match that effort, while Stoke's Tony Pulis will be desperately hoping new boy Kenwyne Jones makes a swift recovery, with the Potters appearing to lack firepower up front.
The summer signing also hit the post in the second half before a furious row erupted between opposing managers Steve Bruce and Rafael Benitez, who clashed after Sunderland striker Kenwyne Jones had extensive treatment on the pitch before being stretchered off, like teammate Lee Cattermole was earlier.
However, after the break, the diminutive winger imposed his authority on the game and after delivering a delightful cross for Huth, he found Dean Whitehead and Kenwyne Jones with two more accurate centres and although they did not lead to goals, Stoke ran out comfortable winners thanks to Marc Wilson's late strike.
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