The US Treasury Department announced the measures against 13 people and a dozen companies on Friday.
President Donald Trump tweeted earlier: "Iran is playing with fire - they don't appreciate how 'kind' President Obama was to them. Not me!"
But Iran has said it will not yield to "useless" American threats from "an inexperienced person".
John Smith, the Treasury Department's acting sanctions chief, said in a statement: "Iran's continued support for terrorism and development of its ballistic missile programme poses a threat to the region, to our partners worldwide and to the United States."
US approach changes - Kim Ghattas, BBC News
President Obama may have sanctioned Iran for its missile test a year ago as well, but President Trump's sanctions come in a very different context and from a very different team.
This administration is filled with officials whose sole focus is Iran, or even hold a grudge against it, like the Defence Secretary James Mattis.
Mr Obama focused on fostering a tone that wouldn't jeopardise the Islamic Republic's commitment to the nuclear deal. He rarely referred to Iran's paramilitary activities in the region.
But the Treasury Department's mention on Friday of "Iran's malign activity abroad" was a reference to Iranian support for Shia militias and involvement in countries such as Syria and Iraq.
There may be still be echoes of Obama's policies here, but the whole framework of the approach has changed and Mr Trump and his team are signalling clearly they want to cut Iran to size.
Some of the newly sanctioned groups are based in the United Arab Emirates, Lebanon and China, and include members of the Islamic Republic's Revolutionary Guards Corps.
Image captionIran 'put on notice', says White House
They are the first Iran sanctions of Mr Trump's new presidency, and come a day after he said "nothing is off the table" in dealing with the country.
Oil prices rose on Friday morning, as markets factored in the announcement.
This week, the US national security adviser, Michael Flynn, said the administration was putting Iran "on notice" for its medium-range missile test.
The White House said the launch had violated a UN Security Council resolution proscribing missiles that could carry a nuclear device.
But Iran's Foreign Minister Mohammad Javad Zarif tweeted on Friday that the Islamic Republic was unmoved by US threats.
"Will never initiate war, but we can only rely on our own means of defence," Mr Zarif wrote.
The Republican Speaker of the House of Representatives, Paul Ryan, applauded Mr Trump on his administration's "swift and decisive response", in a statement on Twitter.
More than a dozen US senators from both main parties wrote on Thursday to the president, urging "full enforcement of existing sanctions and the imposition of additional sanctions on Iran".
Meanwhile, Saudi Arabia's senior military advisor, Brigadier General Ahmed al Assiri, told the BBC it was time to change Iran's behaviour in the region.
Speaking in the Saudi capital Riyadh, he said Tehran's involvement in neighbouring countries such as Iraq, Syria and Yemen must be stopped.
Gen Assiri, who advises the Saudi defence minister, said Iran must be "brought back to its borders".
Iran - long an arch-foe of Saudi Arabia - says its presence across the region is always at the request of the governments.
The sanctions came as the US moved a Navy destroyer closer to the coast of Yemen to guard waterways from the Iran-aligned Houthi militia, Reuters news agency reports.
Also on Friday, Iran announced a ban against US wrestlers from competing at a championship in the western Iranian city of Kermanshah later this month.
The ban was in response to Mr Trump's executive order temporarily barring travel to the US for Iranian citizens, as well as citizens of six other majority-Muslim countries.
