The cultural centre, which rises from the landscape in the Azerbaijani capital, is also the first architecture project to be lauded.
"We're absolutely delighted to receive the Design of the Year Award," said Hadid. "The surface of the Heydar Aliyev Center's external plaza rises and folds to define a sequence of public event spaces within; welcoming, embracing and directing visitors throughout the building."
"It's an architectural landscape where concepts of seamless spatial flow are made real – creating a whole new kind of civic space for the city," she added.
The building was designed to host exhibitions, concerts and other cultural activities beneath the folds of glass-fibre-reinforced concrete panels.
"It's beautiful, it's inspiring, it's the clear vision of a singular genius and we thought it was a remarkable piece of work," said writer, broadcaster and jury chair Ekow Eshun.
Speaking to Dezeen last week, Hadid said that she felt the centre was the most important of her recent projects – along with her Aquatics Centre for the London 2012 Olympics.
A shortlist of 70 projects were nominated in categories of architecture, digital, fashion, furniture, graphics, product and transport. All shortlisted projects are on show at the Design Museum until 25 August.
Dezeen is media partner for Designs of the Year 2014, which also included a Social Vote decided by the public – who picked Dave Hakkens' Phoneblocks modular mobile phone concept as their favourite. See more Designs of the Year 2014 projects »