Saturday's US commando raid to free 33-year-old American Luke Somers came two days after the kidnappers had issued a video in which they threatened to kill him within 72 hours.
But it also came a day before the South African hostage, 57-year-old Pierre Korkie, had been due to be released under a negotiated deal.
Calling the murders "barbaric," Obama said he had authorised the rescue attempt because the video and other information "indicated that Luke's life was in imminent danger".
The South African government said Sunday it was "deeply saddened" by the killing of Korkie.
It said in a statement that it had undertaken "numerous initiatives to help secure Mr Korkie's release."
A top US lawmaker said Sunday he agreed with the president's decision to carry out the raid.
"The intelligence was good," Mike Rogers, chairman of the House Intelligence Committee, told CNN television.
US Defence Secretary Chuck Hagel said there had been "compelling reasons to believe Somers's life was in imminent danger".
British-born Somers had worked as a freelance photographer for the BBC and spent time at Yemeni newspapers before being abducted in Sanaa in September 2013.
Korkie and his wife Yolande, who had worked as teachers in Yemen for four years, were kidnapped by Al-Qaeda in the Arabian Peninsula (AQAP) in May 2013.
She was released in January and the Gift of Givers charity said logistical arrangements had already been put in place to fly Korkie out of Yemen on Sunday.
They made their way to the Al-Qaeda hideout by foot, but were discovered about 100 metres away.
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