WASHINGTON—The Obama administration said Tuesday it will formally shift authority for much of the internet’s governance to a nonprofit multi-stakeholder entity on Oct. 1, a move likely to spark a backlash from parts of Congress.

The administration—as well as many in the high-tech community—regard the long-planned move as necessary to maintain international support for the internet and prevent a fracturing of its governance. They say transferring authority for the internet’s domain-name system to the Internet Corporation for...