Tehran has acknowledged that it supplied Moscow with drones but said they were sent before the war in Ukraine started in February.
The US said Russia has used drones from Iran to attack cities and energy infrastructure in Ukraine.
“Iranian-supplied weapons threaten the entire region. Today Iran’s nuclear programme is more advanced than ever before, and the regime has resorted to selling Russia the armed drones that are killing civilians in Ukraine,” Mr Cleverly said at the annual International Institute of Strategic Studies Manama Dialogue.
The forum in Bahrain's capital is organised by the IISS, a think tank in London.
“As their people demonstrate against decades of oppression, Iran’s rulers are spreading bloodshed and destruction as far away as Kyiv,” Mr Cleverly said.
He said the UK would work with partners in the region to ensure Tehran never developed a nuclear weapon and to highlight the effects Russia’s invasion of Ukraine had food security across the region.
“Britain is determined to work alongside our friends to counter the Iranian threat, interdict the smuggling of conventional arms and prevent the regime from acquiring a nuclear weapons capability,” he said.
Iranian military entities and industries are already under heavy US sanctions over Tehran's nuclear programme.
This week, Iran's mission to the UN said the country was ready to meet Ukraine at the level of technical experts and to investigate claims about the origin of Russian drones.
In recent months, Washington has looked to increase pressure on Moscow over its invasion, which has reduced cities to rubble and killed and wounded thousands.
Mr Cleverly has described the invasion of Ukraine as a “flagrant breach” of the principles of sovereignty and territorial integrity that is “heaping misery” on millions of Syrians and Yemenis by driving up food prices.
“No country is immune from the turmoil he has brought to world energy markets or the damage he has caused to global food security,” he said.
“[Russian President Vladimir] Putin’s war is inflicting yet more suffering on Syrians and Yemenis, who were already enduring the privations of humanitarian emergency, and ordinary Lebanese, caught up in economic crisis.”
Mr Cleverly said Russia was losing the war in Ukraine.
"It should be dawning on other regimes who might have been tempted to behave similarly that most of the world is determined to ensure that aggression does not pay," he said.
He called on the international community to work together to ensure Russia is faced with collective sanctions to stop the conflict.
"We have learnt that sanctions are most effective when they are internationally co-ordinated," he said.
The war in Ukraine is a "conflict that attacks the foundations of peace and stability that we have enjoyed for decades", he said.