The publicly-listed company had opposed an £8.9bn takeover offer but agreed to the new deal after Schaeffler upped its bid and limited itself to a minority stake.
Continental has insisted that as part of the deal, it will not be broken up and that it will keep its stock market listing.
Continental's chief executive Manfred Wennemer - who had vehemently opposed the plan - is to quit later this month.
He had been angry at Schaeffler secretly building up a 36% stake in the company.
Privately-owned Schaeffler makes parts for car and plane systems and specialises in the manufacture of ball bearings.