Cisco, based in San Jose, builds the routing and switching gear that has powered the growth of the Internet and telecommunications systems worldwide, and the company's revenue totaled $47 billion in its fiscal year ended last July 26.
Chambers, 65, has been Cisco's chief executive for the last 20 years. He will remain chairman, and assume the role of executive chairman July 26 when Robbins becomes CEO.
Robbins, who joined Cisco in 1997, most recently has been the company's senior vice president of worldwide operations.
"This is the perfect time for Chuck Robbins to become Cisco's next chief executive," Chambers said in a statement.
"Chuck knows every Cisco segment, technology area and geography, and will move the company forward with the speed required to capitalize on the opportunities in front of us," Chambers said.
Robbins also will face headwinds that have buffeted Cisco, including increasing competition, especially in the telecommunications sector.
"Humbled and honored," Robbins said on Twitter after Cisco's announcement.
Cisco's stock, a component of the Dow Jones Industrial Average, was up 15 cents to $29.28 a share in morning trading Monday.
The shares were among the high-fliers during the stock market's tech bubble at the turn of the century, trading above $70 a share on a split-adjusted basis in 2000 before the bubble collapsed.