The broadcaster said she was “at home with her family by her side” and died on Tuesday after a short illness – which was not related to coronavirus.
Bernicoff appeared on the reality TV show with her late husband Leon.
“As the first couple to be cast for Gogglebox back in 2013, June and her husband Leon were a huge part of the programme’s success,” Channel 4 said.
“Their warmth, wit and contrasting personalities endeared them to the nation during the course of the first 10 series.”
Writing on Twitter, Gogglebox producer Tania Alexander said: “June and Leon were the Gogglebox originals & a huge part of the show’s success. I adored them both. Big kiss June darling.”
The Bafta-winning show features people sitting in their living rooms watching and reacting to television programmes.
June left the show after Leon’s death in December 2017 at the age of 83. She went on to write her first book about their 60-year love affair, titled Leon And June: Our Story.
“Our thoughts are with June’s family – Helen and Ian, Julie and Marc, and her beloved grandchildren Frances, Sam and Faye,” Channel 4’s statement said.