"Following the expiration of the public offer's term, Daimler and Rolls-Royce together hold 59.87 percent of the shares in Tognum via their joint company Engine Holding," a statement said.
At €26 per share, the deal cost a little more than €2 billion and the offer has been extended until June 20 so Engine Holding can bolster its stake further.
The takeover will create a global leader in the "off-highway" sector of marine, energy, defence and industrial applications.
Daimler and Rolls-Royce last month improved their initial offer by 8.3 percent to win the backing of Tognum management.
Daimler already owned 28.4 percent of Tognum, which employs some 8,700 people around the world, and has a long history with the company.
Having owned it for nearly 50 years, Daimler sold Tognum in 2005 to the EQT investment fund for €1.6 billion, only to later buy back around 22 percent in April 2008 for €585 million.