Operators shut two lifts for safety reasons, stranding some visitors on an observation deck for half an hour.
Tens of thousands of people had flocked to the Japanese capital's newest attraction.
At 634m, the Skytree is twice the height of the Eiffel Tower. The Burj Khalifa in Dubai remains the world's tallest man-made structure at 828m.
The Skytree, which took three-and-a-half years to build, offers broadcasting services across the Tokyo area.
The main attractions of the needle-shaped structure are two observation decks at 350m and 450m above ground.
Wind was not the only problem on opening day - rain also dampened proceedings, spoiling what should have been far-reaching views over the city.
"I can't see the view, but it was exciting," said Ayumi Nakazawa, the first official visitor to the observation deck, AFP news agency reported.
The Skytree stands about twice the height of Tokyo Tower, the city's landmark tower since the 1960s.
Source: Council on Tall Buildings and Urban Habitat
Major broadcasters, including Japan's public broadcaster NHK, will begin using it for transmissions from next year, reports said.
The first observation deck of the Skytree can accommodate up to 2,000 people and the second deck up to 900, said local media reports.
The tower withstood damage from the devastating earthquake that hit Japan in March 2011 during its construction.
The disaster pushed back building efforts by two months, but no-one was hurt and construction was resumed.
The tower is now viewed as a testament to Japan's earthquake-resistant building technology, as well as a symbol of resilience, local media reports said.
But it seems it is not immune to the effects of everyday elements, such as strong wind.
High winds have marred the opening to the public of the world's tallest broadcasting tower, Tokyo Skytree.