The world's biggest spirits company, which makes Tanqueray gin and Don Julio tequila, said Crew would take up her new role and also join the board on 1 July.
Crew, an industry veteran who became COO last year, had been president of Diageo North America and Global Supply from 2020.
"Crew was the most likely CEO successor, so no surprise there," said Tineke Frikkee, a fund manager at Diageo investor Waverton Asset Management. "She is very experienced in Diageo’s largest market, the US."
"The timing of succession is always difficult to predict but with Ivan Menezes aged 63, it was imminent. I expect Diageo strategy to remain unchanged."
Crew was previously CEO of tobacco company Reynolds American, where she previously served as COO.
Menezes joined Diageo after the merger of Guinness plc and Grand Metropolitan plc in 1997 and has held a number of senior positions in the business.
News by Reuters edited by Donna Ahern, Checkout. For more drinks stories click here. Click subscribe to sign up for the Checkout print edition. Sign up to subscribe to our weekly newsletter Retail Intelligence.