It was the first visit to Tehran by a Western foreign minister since the United States withdrew from the multi-nation nuclear deal in May.
Hunt met his counterpart, Foreign Minister Mohammad Javad Zarif, with the two ministers discussing plans to keep trade flowing between Iran and the UK in spite of renewed US sanctions, according to the semi-official ISNA news agency.
But Hunt was particularly focused on the conflict in Yemen, where Iran is accused of supplying weapons to Houthi rebels, and the jailing of a British mother in Tehran.
"We are very, very keen to move towards peace in Yemen. That's our number one priority at the moment," he told the BBC.
"But also we have the case of Nazanin Zaghari-Ratcliffe and other dual nationals here who are in prison and shouldn't be. We want to get them home."
Zaghari-Ratcliffe is serving a five-year jail sentence for alleged sedition, with Hunt under pressure to free the British mother.
"I arrive in Iran with a clear message for the country's leaders: putting innocent people in prison cannot and must not be used as a tool of diplomatic leverage," Hunt said in a statement before leaving London.
Britain is determined to keep Iran in the nuclear deal, technically known as the Joint Comprehensive Plan of Action (JCPOA).
"The Iran nuclear deal remains a vital component of stability in the Middle East by eliminating the threat of a nuclearised Iran," Hunt said in the statement.
"It needs 100-percent compliance though to survive. We will stick to our side of the bargain as long as Iran does.
"But we also need to see an end to destabilising activity by Iran in the rest of the region if we are going to tackle the root causes of the challenges the region faces."
Iran's patience has been tested by Europe's slow progress in finding ways to work around US sanctions, Tehran has claimed.
"If Europe thinks that the JCPOA is important for its sovereignty, security and credibility, it must be ready to pay for it," deputy foreign minister Abbas Aragchi told ISNA.
"The price of losing the JCPOA is greater for Europe than the US," he added.
President Donald Trump announced the US was withdrawing from the JCPOA in May, describing it as the "worst deal ever", with sanctions planned for Tehran for its missile building programme.
