The precinct, which has two levels and 103 shop units, was put on the market earlier this year for £50m.
It is currently in the middle of a £30m revamp which will include a new Debenhams anchor store, nine larger units, a new atrium and a reconstructed retail mall.
Benson Elliot said it was an important purchase for its investment programme.
"The Mander Centre is a well-located, robust business with a solid profile and a loyal customer base," said managing partner Marc Mogull.
He described the purchase as "a significant opportunity to rejuvenate the Mander Centre, making it once again the centrepiece of a thriving Wolverhampton city centre".
Councillor Peter Bilson, Wolverhampton City Council's cabinet member for economic regeneration and prosperity, said: "These are momentous times for our city, with a level of activity we haven't known since the 1960s.
"It's a clear sign of the progress Wolverhampton's making that a major private equity firm like Benson Elliot should want a stake in its future. They have great plans for the Mander Centre - and on behalf of the council and all its partners, I'd like to welcome them to a city that is really making it happen."
The Mander shopping centre in Wolverhampton has been sold for £59m to property investment firm Benson Elliot.