Bahah was sworn in in front of exiled President Abedrabbo Mansour Hadi, a day after his appointment, Saudi official state news agency SPA reported.
It said several Yemeni ministers, officials and political leaders attended the ceremony.
The Gulf Cooperation Council welcomed Bahah's appointment as an important step towards helping Hadi's efforts to "restore security and stability in Yemen."
The bloc which groups Saudi Arabia with Bahrain, Kuwait, Oman, Qatar and the United Arab Emirates pledged to "continue to support all efforts" to push for a political solution in Yemen.
Both Hadi and Bahah had been placed under house arrest in Sanaa by Shiite Huthi rebels who overran the capital in September and tightened their grip on power after the two leaders tendered their resignations in January.
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But while Hadi escaped in February to resurface in Aden, Bahah reached a deal with the Iran-backed Shiite militia to end his house arrest in March.
He headed to Hadramawt, his native province in the southeast.
On March 24, Bahah reportedly travelled to New York where his family lives, as the rebels tightened the noose on Aden, two days before Saudi Arabia mounted an air campaign in support of Hadi.
He flew earlier this month to Riyadh where Hadi has taken refuge as fighting raged in Aden.
Bahah studied administration and finance in his home country as well as India. He held the post of oil minister between 2006 and 2008, before serving as ambassador to Canada until 2014.
He returned to the oil ministry for a few months, before briefly becoming Yemen's envoy to the United Nations in New York, and Bahah was then appointed prime minister in October 2014.
Deeply-tribal Yemen, home to a deadly Al-Qaeda franchise, is the poorest Arabian Peninsula state and has seen many conflicts. Violence has escalated since the Huthis overran Sanaa.