Mr Aingimea, a second term MP, defeated David Adeang by 12 votes to 6.
He succeeds Baron Waqa, who lost his seat in Saturday's general election.
Former president Marcus Stephen, is the new Speaker of Parliament.
He won by 12 votes to 7 over Shadlog Bernicke when the new Parliament voted today.
Mr Aingimea is a former secretary of justice and served as an assistant minister in various portfolios in the previous government.