The plan by Raiders owner Mark Davis, who has been the driving force behind the relocation effort, won the support of 31 of the league's 32 owners, with only the Miami Dolphins' ownership dissenting.
"My father used to say that the greatness of the Raiders was in its future," Davis said, referring to Al Davis, from whom he inherited the team in 2011.
"The opportunity to build a world-class stadium in the entertainment capital of the world will give us the opportunity to achieve that greatness," he added following the vote.
The Raiders will play the 2017, 2018 and possibly 2019 seasons in Oakland before kicking off the 2020 season in Las Vegas, Davis said.
He acknowledged some fans in Oakland will be disappointed and even angry at the decision, but said that frustration should be directed at him and not the team's coaches or players.
He said that in the meantime, his goal was to "bring a championship back to Oakland."
The relocation plan appeared to be all but dead after casino mogul Sheldon Adelson and later Goldman Sachs earlier this year changed their minds about helping to finance the stadium construction.
Adelson had pledged up to $650 million toward construction of the domed stadium but pulled his support in January after the team presented a lease proposal without his knowledge.
Read the full article on eurosport.co.uk: NFL owners approve Raiders' move to Las Vegas from Oakland