Mr Quang, 59, has been head of the ministry of public security, which has been the focus of Western criticism of the nation's human rights record.
He had been nominated for the largely ceremonial role at January's communist party conference.
One of his first tasks will be to welcome visiting US President Barack Obama next month.
Mr Quang won 91.5% of a rubber-stamp vote in parliament on Saturday.
"I sincerely thank the National Assembly for electing me," Mr Quang said as he was sworn in. He is the first police general to fill the post.
The Communist Party in January re-elected Nguyen Phu Trong, 71, in the leading role of general secretary for a second term.
His re-election came after reformist Prime Minister Nguyen Tan Dung withdrew his candidacy.
The National Assembly will vote next week on a new prime minister - set to be Deputy PM Nguyen Xuan Phuc,
Earlier in the week, the assembly elected its first woman Speaker - Nguyen Thi Kim Ngan.
Analysts believe one-party Vietnam will continue with economic reforms, but steer clear of major political changes.
Vietnam's police chief, Tran Dai Quang, has been sworn in as the communist country's president.