Ballmer said Lawrence Frank, the team's executive vice president, will assume responsibility for basketball operations. Frank also will oversee general manager Dave Wohl.
Both Rivers, 55, and Frank, 46, will report directly to the owner.
"Doc knows how to win championships," Ballmer said in a news release. "That is what we prioritize, and that is what Doc will focus on. He is key to integrating our new players with our returning players and taking us to new heights on the court."
Rivers will continue to have a strong voice in personnel and organizational matters and will partner with Frank, Ballmer later told ESPN on Friday.
"I've owned the team for three years now, and I really better understand what an owner's responsibility is -- and it turns out that running a franchise and coaching are two enormous and different jobs," Ballmer told ESPN. "The notion that one person can fairly focus on them and give them all the attention they need isn't the case. To be as good as we can be, to be a championship franchise, we need two functioning strong people building teams out beneath them. There needs to be a healthy discussion and debate with two strong, independent-minded people.
"There are different relationships that a player needs to have with the coach and the front office. Doc put Lawrence in charge of the non-coaching aspects of the front office last year, and he's done a fantastic job. I want each of them to dig in and do what they do best. Lawrence has come on so strong in that role, and that has helped us go down this path."
"I want to have a world-class front office that identifies college and pro talent, that works with the agents and that has a long-range strategy that is based upon thinking through the analytics and options available to us," Ballmer told ESPN.
Rivers and Ballmer promoted Frank to oversee the day-to-day operations of the front office a year ago. Frank is a two-time head coach with the New Jersey (now Brooklyn) Nets and the Detroit Pistons.
Rivers, who left the Boston Celtics to join the Clippers as the franchise's top basketball executive and coach four years ago, inherited a depleted front office in the aftermath of former owner Donald Sterling's racially-charged audio tape scandal.
Ballmer purchased the Clippers for $2 billion and ultimately signed Rivers to a five-year, $55 million contract extension to continue as president and coach in 2014.
In four years as Clippers coach, Rivers has a 217-111 regular-season record, including 18-22 in the playoffs. He reached the NBA Finals twice as Celtics coach, winning the 2008 championship.