Kenneth Chenault, the head of American Express Co. and one of the country’s most prominent African-American corporate leaders, will step down as chairman and chief executive Feb. 1, capping a 16-year run at the iconic card company as it grapples with a new wave of competition.

The 66-year-old executive will be succeeded as CEO by Stephen Squeri, a three-decade AmEx veteran who previously ran its division in charge of corporate cards. As vice chairman since 2015, Mr. Squeri, 58, had spent more time meeting with shareholders,...