The success follows lobbying from the building's architect, Lord Rogers, and senior director Ivan Harbour to ensure that the scheme went ahead. Lord Rogers met first secretary Rhodri Morgan and made a presentation to assembly members casting doubt on alternatives.
In a free vote, assembly members were asked to vote for one of two options: to go ahead with the original design or doing nothing.
Members overwhelmingly backed the Rogers design, voting 48 for and nine against.
Morgan's alternative, known as the "lean to", has been priced at £13.5m. Lord Rogers claimed it would be 50% more expensive per square yard.