Athletes from 87 nations paraded before 40,000 people in the Fisht Stadium before president Vladimir Putin declared the Games open.
The Olympic flame was lit by Russian former triple gold medallists Vladislav Tretiak and Irina Rodnina as fireworks illuminated the night sky.
At a cost of £30bn, the Sochi Games is the most expensive Olympics in history.
The build-up to the Winter Olympics has been overshadowed by security fears, allegations of corruption, complaints from unpaid construction workers and concern about the rights of sexual minorities in Russia.
International Olympic Committee president Thomas Bach used his welcome speech at the ceremony to promote Olympic values of diversity and non-discrimination.
"The Olympics are about building bridges to bring people together," Bach said. "They are not about erecting walls to keep people apart. Embrace human diversity and unity."
US President Barack Obama and British Prime Minister David Cameron stayed away from the ceremony, which was attended by 44 heads of state.
Putin, the driving force behind the Games, avoided making a political statement as he stuck to protocol by using a single sentence to declare the Games open.
Spectators inside the stadium and television viewers worldwide were treated to a glittering show lasting nearly three hours and featuring 22 tonnes of fireworks.
There was a minor glitch early on when one of five giant snowflakes failed to open up to form an Olympic ring, leaving the Games' symbol incomplete.
However, the ceremony director Konstantin Ernst played down the incident, saying: "Zen buddhists have a saying that if you have the perfectly polished ball, leave a nick in it so you can understand just how perfectly it is polished.
"No normal person would get distracted by one snowflake that did not open from the story that is being told over two and-a-half hours."
In a grand finale, tennis star Maria Sharapova, who grew up in Sochi, carried the Olympic flame into the stadium before it passed through the hands of pole vault world record holder Yelena Isinbayeva, wrestler Aleksandr Karelin and gymnast Alina Kabaeva,
It was carried out of the arena by former figure skater Rodnina and ice hockey player Tretiak, who lit the cauldron in the middle of the Olympic park while Stravinksy's "Firebird Suite" filled the stadium.
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