The hostages, Jessica Buchanan and Poul Thisted, were seized in October after they visited humanitarian projects in northern Somalia, said the Danish Refugee Council, the agency for which they worked.
Both are unharmed, the aid group said.
Navy SEALs from the unit that killed al Qaeda leader Osama bin Laden last year in Pakistan were part of the mission, a U.S. official said, without specifying whether any of the same individuals were on both assaults.
The official is not authorized to speak to the media and asked not to be named.
The special forces troops took fire as they fought their way into a compound where the hostages were held, the official said, adding the troops believed that the kidnappers were shooting.
Nine heavily armed gunmen were killed in the strike, Pentagon spokesman Little said, adding that they had explosives nearby.
There were no known survivors among the kidnappers, Little said.
He said SEALs were only part of the special forces team, but would not specify what branches of the military the other troops came from. The SEALs are part of a unit officially called the Naval Special Warfare Development Group, formerly known as SEAL Team Six.
There are conflicting reports of the number of wounded in Somalia. The American assault team did not suffer any injuries, the Pentagon said.
"It just takes your breath away, their capacity and their bravery and their incredible timing," Biden said of the special forces.
The rescued American is "doing well, under the circumstances," her father told CNN on Wednesday.
John Buchanan said the family was "fine -- now," and expressed his thanks to the special forces who rescued her, saying: "We're really grateful."
He declined to comment on her health.
The special forces rushed the hostages out of the compound and onto the helicopters, said the official.
There "is no reason" to believe the kidnappers were acting as part of a larger jihadist group, the official said. The area where the hostages were seized is known as a hub for pirates, rather than an area of Islamic militant activity.
A number of high-profile abductions of foreigners have occurred in Somalia and in Kenya, close to the largely lawless Somali border.
Some of the kidnappings have been blamed on the Somali Islamist militant group Al-Shabaab, while criminals seeking ransoms seem to have carried out others.