Cerberus acquired 80.1 percent of Chrysler from DaimlerChrysler in a $7.4 billion transaction, the Detroit Free Press said.
The deal had been announced in May, but it took the two sides time to complete the details.
Chrysler, which sells cars and trucks under the Chrysler, Dodge and Jeep brands, is undergoing a major restructuring. It lost $680 million last year and $2 billion in the first quarter of this year.
Tom LaSorda is expected to remain as Chrysler chief executive officer and former Chrysler Chief Operating Officer Wolfgang Bernhard is expected to become Chrysler chairman, the Free Press said.