A statement on the Swansea website (www.swanseacity.net) said the investment consortium led by the two businessmen had acquired a "controlling majority" of the shares after lengthy negotiations.
The BBC reported that the 100 million pound ($145.18 million) deal was for 60 percent of the club, who finished last season in 12th position in the league.
Swansea said the shareholding of the club's Supporters' Trust, who have a 21 percent stake and a representative on the board, would remain unaffected.
Several existing shareholders will retain smaller stakes and chairman Huw Jenkins, who remains in charge of the day-to-day running of the club in an executive role, will "maintain an ownership stake".
Swansea said the takeover was expected to be fully sanctioned by the Premier League in the coming weeks.
Levien is the managing general owner of Washington-based Major League Soccer side DC United while Kaplan is executive vice-chairman of the Memphis Grizzlies NBA basketball franchise in the United States.
"Both see the takeover of Swansea City as an exciting opportunity for long-term growth in the club in every area," the club said.
Swansea were promoted to the top flight in 2011 and are managed by Italian Francesco Guidolin.
Their Liberty Stadium, currently owned by the city council, is the second smallest in the Premier League with a capacity of just under 21,000.
The Supporters' Trust said in an earlier statement that they had been assured the purchase of shares by the new owners would be paid for from the consortium's own funds and there was no intention to add new debt to the balance sheet.
"The trust is fully aware of the club's need to progress off and on the pitch," it added.
"We all know that there is considerable scope to improve revenue growth in certain areas and we also understand the need for sustainable investment, especially for projects such as stadium expansion."
The South Wales club are the third in the Premier League to share common ownership with an MLS franchise.
Arsenal's majority shareholder Stan Kroenke owns the Colorado rapids while Manchester City's Abu Dhabi owners also control New York City FC.
($1 = 0.6888 pounds)
(Reporting by Alan Baldwin, editing by Amlan Chakraborty)
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