The US had repeatedly called for the imposition of a renewed embargo on Iran, claiming it was sending Iranian manufactured missiles to Iranian-backed Houthi rebels in Yemen.
It also claimed that these missiles were used to launch an attack on neighbouring Saudi Arabia - which along with several Sunni Arab countries - have been fighting against the Houthis since March 2015, when the Kingdom intervened to push back the rebels and reinstate the government of President Abd-Rabbu Mansour Hadi.
The draft resolution specifically condemned a Houthi missile strike that was targeted at Riyadh in December 2017.
It determined that Iran did not comply with a 2015 UN arms embargo on Yemen as it did not take action in preventing the direct or indirect flow of missiles to the Houthis.
The draft also urged members of the council to take further action against Iran for breaching the arms embargo.
Instead, the UNSC unanimously adopted a Russian-proposed resolution that did not mention Iran.
Although the Russian-proposed one passed unanimously, criticism was traded among the council's members.
Kelley Currie, the US representative to the UN, said the UK-drafted resolution was a "simple, commonsense one" which called out Iran.
Russia's ambassador to the UN, Vassily Nebenzia, said the wording in the British draft would have had "destabilising ramifications" if adopted, and warned about the effect it would have on Yemen and the region as a whole.
"Russia doesn't like the outcome of certain expert panel reports," Jonathan Allen, UK deputy ambassador to the UN, told reporters before the vote was scheduled to take place.
"We'll be urging our colleagues for the good of the United Nations system, to vote in favour of our resolution."
Earlier on Monday, Russian Foreign Minister Sergei Lavrov discussed with his Saudi counterpart Adel al-Jubeir the drafting of a new UNSC resolution on Yemen, Anadolu news agency reported.