The burly 68-year-old populist now has two weeks to form a cabinet and go back to parliament to win final approval for his programme.
“The main task of the new Cabinet will be to continue vital reforms,” he said.
“Joining Nato and the European Union will be major tasks of this government.”
Mr Brazauskas was Communist Party boss during Lithuania's bid for independence from the USSR, then president from 1993 to 1998, when he was replaced by Valdas Adamkus, a former US citizen.
President Adamkus, who nominated him on Friday, was thought to have preferred candidates from the centre-right.
The president warned that he would not stand on the sidelines if he saw any actions taken by the government that jeopardised Lithuania's strategic goals and the reforms needed to achieve them.
Mr Brazauskas has said he will speed up the reforms aimed at securing EU and Nato membership, while also putting greater emphasis on social programmes.
But opponents say the two goals are incompatible.
“The wish to increase social guarantees and to speed up reforms at the same time is impossible,” said Gintaras Steponavicius, a leading member of Liberal Union, the party that headed the outgoing government.
The government crisis was caused by the collapse two weeks ago of a centre-right coalition headed by Prime Minister Rolandas Paksas amid in-fighting over privatisation and other reforms.
During the Paksas government, Lithuania made great strides towards its goal of completing EU entry talks next year so it can join in 2004, when the bloc says it wants to accept the next new members.
Lithuania's chances for EU entry then are seen as good.
Lithuania has also been mentioned as a top candidate to win an invitation to join Nato at the alliance's Prague summit meeting late next year.
In an open vote 84 members of parliament voted for Mr Brazauskas, 45 voted against and three abstained.