The Dundee native flew back from work in Spain to be one of the first through the doors and said it was a “very emotional experience”.
The first V&A museum anywhere in the world outside London, the building designed by Japanese architect Kengo Kuma on the banks of the River Tay is the centrepiece of an ongoing £1 billion regeneration of Dundee’s waterfront.
Mr Cox said: “I think we all agree, those of us who are close to the city and who are close to what is going on here, it’s a very emotional experience.
“Especially for me, I’ve known the city for many years, I was born here, went to school here and had my first theatre roles here, so to return and see what’s happening now - it’s transformed the city.
“It will attract industry and it will attract tourists, it’s the beginning of hopefully a paradigm shift in Dundee and it will put Dundee on the map for the first time.
“It’s an amazing experience. Where it’s situated on the Tay and what Kengo Kuma has done in terms of design - I believe he has said it is his best work.”
Mr Cox said he believed Dundee will come to be defined by the museum and having been involved with the project from an early stage, he promised to attend the opening.
High-school students were among the first of 3,000 members of the public welcomed into the building on Saturday, which is expect to attract 500,000 visitors a year.
Scott Regan, 17, of the city’s St Paul’s Academy, said: “It’s a real privilege being one of the first people in.
“There’s so many people from other schools who could’ve got the chance to be the first one to step in the V&A. It’s really impressive. I can’t believe it’s in Dundee.”
He was joined by Shannon Balfour, 17, from Morgan Academy, Dundee, who said it was an “amazing experience”.
She spoke to Mr Kuma, whose design was chosen from more than 100 entries, and said he is a “really great man”.
“The architecture is amazing and the building inside is just fantastic,” she added.
The Scottish Design Galleries feature 300 exhibits drawn from the V&A’s collections of Scottish design, as well as from museums and private collections across the world.
V&A Dundee director Philip Long said there is a “real sense of optimism” at the opening.
He said: “It’s wonderful to see people coming into the museum and really bringing it alive.”
A two-day festival to celebrate the opening, headlined by Primal Scream on Friday, is expected to attract up to 20,000 people.