Arlene Foster made the call after the Irish tricolor was used in the graphic illustrating differing travel quarantine restrictions in place in the different UK regions during Friday morning’s show.
Mrs Foster’s DUP party branded the incident “disgraceful”.
At the end of the program, presenter Naga Munchetty apologized for the error.
“We would like to apologize for a mistake we made earlier in the program,” she said.
“In a graphic sequence explaining the differing travel quarantine rules around the nations of the UK we mistakenly showed the wrong flag for Northern Ireland.”
Mrs Foster welcomed the apology but urged the BBC’s new director general Tim Davie to establish what went wrong.
She tweeted: “@BBCBreakfast were right to apologize swiftly but the BBC DG needs to investigate the editorial process that allowed the inaccurate graphic to air. Very poor.”
The DUP said one of its MPs, Ian Paisley, had emailed Mr Davie following the error to demand an apology.