Shares of the company rose 1.5% in extended trade to $119.40.
Board member Anthony Melone will take interim charge following Brown's departure, and the company said it will conduct a search for a permanent CEO.
Elliott did not immediately respond to a request for comment on CEO Brown stepping down.
Last month, Elliott said it was ready to nominate directors at Crown Castle and blamed the company's board and management for years of underperformance.
The hedge has also urged the company to review its fiber strategy, including considering a possible sale of the business, optimizing its incentive plan and improving corporate governance.
It is the second time Elliott is publicly pressuring the company after it urged management to rethink its fiber infrastructure strategy and criticized the company's returns in 2020.
Elliott last week also urged U.S. oil refiner Phillips 66 (PSX.N) to revamp its board to boost lagging performance.
