The laws will be fully implemented from April 3 and will allow sodomy, adultery and rape to be punishable by the death penalty, and theft to be punishable by amputation.
"The (Sharia) Law, apart from criminalising and deterring acts that are against the teachings of Islam, also aims to educate, respect and protect the legitimate rights of all individuals, society or nationality of any faiths and race," said a statement from the office of Brunei Prime Minister Hassanal Bolkiah, who is also the sultan.
"Brunei Darussalam is a sovereign Islamic and fully independent country and, like all other independent countries, enforces its own rule of law," the statement said.
Some aspects of the laws will apply to non-Muslims.
The move has drawn widespread criticism, including from politicians in Europe and the United States.
Sebastian Kurz, chancellor for Austria, tweeted."No one should be criminalised based on their sexual orientation or gender identity,"
CNN reported that the US State Department said it was "concerned" by Brunei's decision.
"Some of the punishments in the law appear inconsistent with international human rights obligations. Governments have an obligation to ensure that all people, including LGBTI people, can freely enjoy the universal human rights and fundamental freedoms to which they are entitled," the department said in a statement.
"We strongly oppose human rights violations and abuses against LGBTI persons, including violence, the criminalization of LGBTI status or conduct, and serious forms of discrimination."
Penny Mordaunt, UK secretary of state for international development, said no one should face the death penalty because of who they love.
"Brunei's decision is barbaric and the UK stands with the LGBT+ community and those who defend their rights," Mordaunt said on Twitter.
Former New Zealand Prime Minister Helen Clark said on Twitter: "Hard to comprehend what could be driving such a barbaric move which stands in stark opposition to fundamental #humanrights principles."
Oscar-winning actor George Clooney has called for a boycott of luxury hotels owned by The Brunei Investment Company, such as the Beverly Hills Hotel, the Dorchester in London and the Plaza Athenee in Paris.