Netflix Inc. blew through its forecast for subscriber additions in the September quarter, reassuring investors who were skittish about the streaming giant’s growth trajectory and sending its shares soaring 20% in after-hours trading.

The better-than-expected performance came mainly in international markets, where the company has completed a massive, near-global expansion this year. Netflix is making a big bet that the same mix of edgy original content and library programming that has taken the U.S. media world by storm will...