Robyn Denholm, who has led the electric car company’s board for more than five years, has been criticized for not serving as a check on Mr. Musk.
The company’s board of directors will now decide whether to appeal the decision, change where Tesla is incorporated or negotiate a new pay package.
Shareholders had sued, arguing that Mr. Musk’s compensation — which helped make him the world’s richest person — was excessive.
Shares in Elon Musk’s electric vehicle maker fell sharply after the company delivered lackluster quarterly results and declined to give full-year guidance.
Quarterly earnings soared from a year earlier after the company booked a tax benefit, but increased competition and a price war took a toll, the electric-car maker said.
Elon Musk, the electric car company’s chief executive, said he would “build products outside of Tesla” unless the board raises his stake to 25 percent.
The rental car company blamed the sharp drop in the value of electric vehicles and higher repair costs for its decision to sell 20,000 cars.
Workers seeking a collective agreement from the automaker say they are pushing for their rights, but car owners see them as taking the fight too far.
Federal regulators said the automaker had not done enough to make sure that drivers were paying attention while using Autopilot.
The action began with a mechanics walkout in October, but has spread to include blockades by other unions. Tesla has pushed back through the courts.