SAN FRANCISCO — Netflix’s second-quarter performance followed a familiar script of rapid subscriber growth that has enthralled investors.

The world’s largest Internet video service added 3.3 million worldwide streaming subscribers during the three months to end the period with 65.6 million customers. About 900,000 of the additional subscribers were signed up in the U.S., where Netflix now has 42.3 million customers.

The increases were far better than Netflix’s management projected and represented the biggest gains during the second quarter since the company began streaming video over high-speed connections eight years ago.

By comparison, Netflix picked up 1.7 million subscribers in last year’s second quarter, traditionally a time when the company has more trouble attracting new subscribers and retaining existing customers because people tend to spend more time outside.

The news lifted Netflix’s stock, which already has more than doubled this year. Netflix shares surged $9.27, or 9.5 percent, to $107.40 in extended trading.

The showing provided more evidence of Netflix’s increasing clout in the entertainment industry as a steadily expanding audience embraces the concept of watching video whenever they want on Internet-connected devices.