Hassan Rouhani said on state TV on Monday that his nation is "confronting a bullying power" in a "situation of economic war".
Hassan Rouhani, who also referred to the US as a "bullying power", made the announcement in a meeting with economists that was broadcast on state TV.
The Iranian president said: "America wanted to cut to zero Iran's oil sales ... but we will continue to sell our oil... to break sanctions."
Thousands of Iranian students chanted "death to America" on the eve of the sanctions, which target the country's vital energy and banking sectors.
US flags, an effigy of Uncle Sam and pictures of Donald Trump were set on fire in images broadcast on state TV.
Mr Trump's secretary of state Mike Pompeo has defended the sanctions and heralded them as the "toughest" ever put in place on the country.
Mr Rouhani said in a the meeting on Monday: "I announce that we will proudly bypass your illegal, unjust sanctions because it's against international regulations.
"We are in a situation of economic war, confronting a bullying power. I don't think that in the history of America, someone has entered the White House who is so against law and international conventions."
The Iran nuclear deal was drawn up by the Obama administration and was agreed to by the UK, Russia, France, China, Germany and the EU.
International financial and economic sanctions were lifted in return for Tehran curbing its disputed nuclear activity under UN surveillance.
The US president posted a Game of Thrones-style image on Twitter on Friday, which showed him striding in front of a moody grey background.
Mr Trump had the words "Sanctions are coming, November 5" across his chest.
The font was the same as that used in the hit TV drama and the wording is a riff on one of its famous lines "winter is coming".
The US president, whose voter base backs a tough line on immigration, has said he is a finalising a plan for tougher entry for migrants.
The elections on 6 November are being seen by many as a referendum on Mr Trump's first two years in office.