The broadcaster's media service was one of the most prominent online platforms to use Flash.
Adobe's plug-in has been criticised by some security experts, who said it was a weak point of many sites.
Flash use has been decreasing and the move will provide a further blow, but it will not kill off Adobe's product, said one security expert.
The BBC announced the decision to move towards HTML5 on Tuesday. In a blogpost, it said it was "now confident [it could] achieve the playback quality you'd expect from the BBC without using a third-party plug-in" such as Flash player.
Users have been invited to visit a BBC site where they can set a cookie in their browsers that will allow them to access the HTML5 player when they visit iPlayer in future. However, the Flash version will remain available.
The BBC said it was testing the new player on a range of browsers, including Google Chrome on all platforms, Firefox 41 and Safari on iOS 5 and above.
Others chosen for the test are Opera 32, Internet Explorer 11 and Microsoft Edge on Windows 10 and Blackberry OS 10.3.1.
The BBC added that it was also going  to "move away from the BBC Media Player app on Android devices" with users invited to join a limited beta test in the blogpost.
"We've been working for the last few months to upgrade our existing HTML5 player to make it usable on the desktop as well as mobile and tablet," wrote James East, the BBC's product manager for media playout.
"We'll be working over the coming months to improve the HTML5 player to match the abilities of our Flash player and to refine the functionality and design based on your feedback."
He said that most video programmes would be available on HTML5 on iPlayer immediately but, for those that took longer to become playable, users would be able to use the old player.
Adobe's Flash is one of many plug-ins used to display multimedia content on websites. HTML5, the latest version of the dictionary used by the majority of the worldwide web, incorporates video hosting, where its predecessors did not.
The BBC has joined Netflix and other major firms in adopting HTML5 and information from analytics website Built With shows a downward trend in the number using Flash over the last 12 months.
The iPlayer, which the BBC said received 250 million TV and radio requests in July this year, was one of the most prominent.
However, many of the internet's largest sites still do use Flash, including Amazon, IMDB and the US video site Hulu.
The plug-in has been repeatedly criticised over its security record and, in June this year, Adobe had to release an emergency software patch after a serious vulnerability was found.
But the security expert Chris Green said Flash still had an important role to play and was sometimes unfairly maligned.
"The industry has moved on from trying to shoehorn one thing in, whether that is Flash or Microsoft's Silverlight. It continues to be very effective in delivering rich content into web pages."
He added that media outlets tended to focus on Flash's weaknesses because it was so widely used.
The BBC's iPlayer has been made available using the HTML5 web language, at the expense of Adobe's Flash player.