A frantic US search-and-rescue operation unfolded after the crash of the F-15E Strike Eagle jet on Friday, as Iran also promised a reward for anyone who turned in the "enemy pilot".
A second crew member was rescued earlier.
Trump wrote in a social media post that the aviator is injured but "will be just fine", adding that he took refuge "in the treacherous mountains of Iran".
"This brave Warrior was behind enemy lines in the treacherous mountains of Iran, being hunted down by our enemies, who were getting closer and closer by the hour," Trump wrote.
Trump said the rescue involved "dozens of aircraft" and the US had been monitoring the airman's location "24 hours a day, and diligently planning for his rescue".
The fighter jet was the first US aircraft to have crashed in Iranian territory since the conflict began in late February.
In a separate incident, an A-10 Warthog fighter aircraft was hit and crashed over Kuwait, with the pilot ejecting, US officials said.
Two Black Hawk helicopters that engaged in the search for the missing US air crew member were hit by Iranian fire but made it out of Iranian airspace, the US officials told Reuters.
Earlier, Trump posted on his social media platform that Iran had 48 hours to open the Strait of Hormuz before "all Hell will reign (sic) down on them".
General Ali Abdollahi Aliabadi of Iran's joint military command responded with "the doors of hell will be opened to you" if Iran's infrastructure is attacked, state media reported, with the general threatening all infrastructure used by the US military in the region.
The war, now in its sixth week, began with joint US-Israel strikes on February 28 and has killed thousands, shaken global markets, cut off key shipping routes and spiked fuel prices.
In an apparent move to heap further pressure on Tehran following Trump's latest ultimatum, a senior Israeli defence official said Israel was preparing to attack Iranian energy facilities in the next week, and was awaiting approval from the US.
"We will continue to crush them," Israeli Prime Minister Benjamin Netanyahu said of the Iranian regime on Saturday.
Israel's military confirmed it struck a petrochemical complex in Iran's Mahshahr that Netanyahu said helps to fund the war.
Five people were killed and 170 more were injured, Iranian state media reported, citing a provincial security official.
The Atomic Energy Organisation of Iran said an airstrike hit near its Bushehr nuclear facility, killing a security guard and damaging a support building.
The head of Russia's state nuclear corporation, Rosatom, said 198 workers were being evacuated. It was the fourth time the facility was targeted.
Pakistan's Foreign Ministry spokesperson, Tahir Andrabi, told The Associated Press his government's efforts to broker a ceasefire are "right on track" after Islamabad last week said it would soon host talks between the US and Iran.
Iran's foreign minister, Abbas Araghchi, said Iranian officials "have never refused to go to Islamabad".
Mediators from Pakistan, Turkey and Egypt were working to bring the US and Iran to the negotiating table, according to two regional officials.
Iranian state TV said its military had launched drones at US radar installations and a US-linked aluminium plant in the United Arab Emirates and American military headquarters in Kuwait in retaliation for deadly attacks on Iranian industrial centres.
Iran earlier attacked an Israel-affiliated vessel with a drone in the strait, setting the ship on fire, state media said, citing the commander of the Revolutionary Guards' navy.
Yemen's Iran-aligned Houthis also said on Saturday they attacked Israel using a ballistic missile and drones, adding that the operation was conducted jointly with Iran's Revolutionary Guards, the Iranian army and Lebanon's Hezbollah. The group did not provide evidence of the damage caused.
Israel did not confirm the attack.
The Kuwait Petroleum Corporation said a fire broke out in its Shuwaikh oil sector complex, which houses the oil ministry and KPC headquarters, after a drone attack, the Kuwaiti state news agency reported.
Kuwaiti state media, citing the finance ministry, said an Iranian drone hit an office complex for government ministries, causing significant material damage but no casualties.
With AP
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