Buchanan and Thisted, 60, who were employed by the Danish Refugee Council Demining Group helping to de-mine war-torn Somalia, had been held since October 25 when they were seized by gunmen in the semi-autonomous Galmudug region.
The two freed hostages were unharmed and flown to safety. Reports said they were being treated in Dijibouti, where the US has its only base in Africa.
Obama was told by his top counterterrorism chief John Brennan of the rescue just over two hours before his State of the Union address.
The operation showed "the United States will not tolerate the abduction of our people, and will spare no effort to secure the safety of our citizens," the president said in a statement.
"This is yet another message to the world that the United States of America will stand strongly against any threats to our people."
Pentagon officials said a convergence of factors led to Monday's decision to launch the operation, but also refused to confirm many specific details, including what Buchanan is ailing from.
"There was a window of opportunity for mission success and we had information that suggested that one of the hostages, Ms Buchanan, might have a very serious medical condition that could be life-threatening," said Pentagon spokesman George Little.
Navy Captain John Kirby, another Pentagon spokesman, said the military had reason to believe Buchanan's condition was pre-existing when she was taken hostage "and also we had reason to believe that it was getting worse."
The commando raid on Somali soil is one of the first to have been officially acknowledged by the US, which admits to flying surveillance drones but rarely comments on reported missile strikes against Al-Qaeda-linked militants there.
The Horn of Africa nation has been without an effective government since it was plunged into a civil war two decades ago, allowing militias to flourish, with Islamist insurgents and pirate gangs ruling mini-fiefdoms.
Memories are also still raw in the United States of shocking scenes in 1993 when two US Black Hawk choppers were shot down and the bodies of 18 US soldiers dragged through Mogadishu's streets.
Obama appeared to have hinted at the mission when he greeted defence Secretary Leon Panetta in the Congress late on Tuesday and told him: "Good job tonight. Good job tonight."
Straight after his address, Obama returned to the White House and with his wife, Michelle, at his side, called John Buchanan to tell him his daughter was safe. The White House released a photo of Obama during the phone call.
General Carter Ham, head of US Africa Command praised the "courageous, competent and committed" men who carried out the raid, at least 100 kilometres inside Somalia.
But the Pentagon would not confirm reports that the same unit of Navy SEALs who killed Al-Qaeda leader Osama bin Laden in a raid on his Pakistan compound in May were involved.
Burnishing his credentials as the nation's commander-in-chief, Obama hailed bin Laden's death and told lawmakers that one of his proudest possessions was the SEAL flag carried on the raid.
Danish Foreign Minister Villy Soevndal said meanwhile his country had been kept in the loop throughout as the drama unfolded in remote scrubland in the notorious pirate region of Adado in central Somalia.
Witnesses said US forces took control of Galkayo airport during the raid with several support aircraft landing there, but it received mixed reactions, with some unhappy at the US military action.
"I hate the pirates, but I can't support Hollywood style killing like this, it would be better if the pirates were arrested, face justice and sentenced to 20 years," said taxi-driver Abdullahi Mohamed Jama.
- AFP
