The corporation's director-general Lord Hall was forced on Monday night to overturn a decision made by the BBC's Executive Complaints Unit (ECU), which had ruled that she had breached editorial guidelines.
During an interview with LBC Radio, the Prime Minister said he had "never heard of her" when asked by host Nick Ferrari if he thought she had indeed breached the corporation's editorial guidelines "by calling Donald Trump a racist".
Mr Johnson said: "I'm so sorry, I just don't, I've never heard of her."
Pressed by Ferrari again, who asked "You don't know the story? You've never heard of Naga Munchetty?", the Conservative Leader replied: "No, I'm sorry."
The radio host then said the "row over racism", to which Mr Johnson replied: "You're going have to forgive me. I should know about Naga Munchetty, but does anybody know about Naga Munchetty. Hands up? You know about Naga Munchetty?"
Lord Hall said in correspondence to BBC staff that the ECU ruling had sparked "an important debate about racism" and the interpretation thereof.
He added: "Racism is racism and the BBC is not impartial on the topic. There was never a finding against Naga for what she said about the President's tweet."
After reviewing the ECU decision himself, he said: "I don't think Naga's words were sufficient to merit a partial uphold of the complaint around the comments she made.
"There was never any sanction against Naga and I hope this step makes that absolutely clear. She is an exceptional journalist and presenter and I am proud that she works for the BBC."
The BBC faced a backlash in the days after the ruling and a number of prominent black and Asian journalists and broadcasters called for the decision to be reversed.
Munchetty has not yet commented and is due to return to present BBC Breakfast on Thursday alongside Charlie Stayt.