The internet shopping giant stepped in after Google had previously been in talks to buy the website but no deal materialised.
Twitch, based in San Francisco, has grown rapidly amid the increasing popularity of video gaming as an online spectator sport.
Its network had 55 million unique users in July, up from 20 million in 2012, with most visiting its website to watch other people play live or view recorded games. Around one million broadcasters post content to the site.
The games streams are interspersed with advertising and it is a lucrative market, with the value of digital video commercials expected to reach around £3.6 billion this year.