Jonny Morris, a Labour councillor in Plymouth, directed the salute towards opposing members during a debate on the rise of council tax.
Speaking to the BBC after the debate, Councillor Morris said he was "very angry at the closing down of debate".
He apologised after other councillors said the action was "an utter disgrace".
More on the councillor making a Nazi salute, and other Devon news
The budget meeting saw councillors narrowly vote in favour of increasing council tax bills by 4.49 per cent.
Mr Morris added: "I let that anger get the better of me and made an inappropriate and offensive gesture. I apologise unreservedly."
A councillor has "apologised unreservedly" after making a Nazi salute during a council debate.