The performance, which will air on Saturday November 3 at 6.30pm, will also be the first time a UK opera house has offered a complete production online.
The broadband video streams will be available on the BBC Radio 3 website for seven days, starting on the same evening as the broadcast of the opera on-air.
Music-lovers will have unprecedented access to ENO's 'Carmen'. To view it audiences can go directly to or click through from Eno's .
The Eno production of 'Carmen', part of the Sky Arts season, is directed by filmmaker Sally Potter, who has previously directed 'Orlando' and 'The Tango Lesson'. She will also direct the video streaming.
'Carmen Online' is a co-production between Eno and BBC Radio 3, in association with Adventure Pictures.
Roger Wright, controller of BBC Radio 3, said: "This is an exciting opportunity for BBC Radio 3 and Eno to bring a new opera production to the widest possible audience and we are delighted to extend our relationship with Eno in this way.
"Using technology to give audiences a richer experience is something we want to continue to explore further in the future."
John Berry, artistic director at Eno, said: "Through this streaming many people across Britain will have the opportunity to sample Eno's bold and adventurous work.
"The web is giving Eno new opportunities for engaging with a wider audience and this project expands on the exciting Web 2.0 content we have created around our production of 'Carmen'.
"We are delighted to be entering a co-production partnership with BBC Radio 3 and we hope it will lead to more exciting projects in the future."
Carmen extends Eno's ambition to create online content for people who want to engage with opera, music, theatre and film.
Earlier this year the company launched Eno Interactive using user-generated content and social networking to create a community that could interact with Eno's work and each other.
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