A club spokesman confirmed the two resignations.
"Bob and the McCourts have determined that Bob will not be with the ballclub in the future and Bob terminated his employment as a result of their discussions," said senior vice president Derrick Hall. "It was Kris' choice to leave and this gives the McCourts the opportunity to determine who will fill the role and what it will entail."
Rone was the fourth-highest ranking women in Major League Baseball.
Owners Frank and Jamie McCourt are apparently in the process of hiring Mike Dee, the Red Sox's executive vice president of business affairs, to succeed Graziano, in the president's role since 1998, as head of day-to-day operations.
Dee formerly was the marketing director for the San Diego Padres under Larry Lucchino, now Boston's president and chief executive. Dee is said to be among Lucchino's top advisors, so Boston could seek compensation from the Dodgers.
Hall declined comment on the possible hiring but added, "this particular position is currently being evaluated and addressed."
Graziano joined the Dodgers in 1986 as the director of financial projects, hired by Peter O'Malley. He became a vice president in 1987, and took over day-to-day operation of the business side of the franchise when he was named executive vice president in 1997.
Fired after the 1999 season by News Corp., Graziano was rehired when Bob Daly took over operation of the franchise before the start of the 2000 season.
Dodgers' president Graziano resigns
LA Dodgers Insider Top Stories
Kershaw VS. Hendricks NLCS Game 6Kershaw VS. Hendricks NLCS Game 6
LA Dodgers InsiderSaturday at 1:01 PM