Swansea chairman Huw Jenkins said the Welsh club were "extremely pleased" with the outcome of the Premier League meeting.
The Swansea City Supporters' Trust, meanwhile, will retain a shareholding of just over 21 per cent.
In a joint statement, Kaplan and Levien said: "We are delighted that the Premier League has ratified our deal to buy a controlling interest in the club.
"We had a very positive meeting yesterday, and we appreciate the fact the Premier League has moved so swiftly to give us the green light to move forward.
"There are still a few minor loose ends to tie up on the deal, which we envisage will be completed before the end of the month.
"In the meantime, we are both extremely excited to be part of a new era for Swansea City and working with Huw Jenkins and the Supporters' Trust in taking the club forward."
Jenkins added: "We are all extremely pleased with the outcome of the Premier League meeting.
"It will now enable us to move forward fully focussed and put everything in place for the start of another very important Premier League season.
"Everyone at the club is looking forward to working together with Steve and Jason for what will be a new and exciting chapter in the club's proud history."
A consortium fronted by Kaplan and Levien has been in talks over acquiring control of Swansea since last year, with news of the proposed change of ownership emerging in early April this year.
It is understood that the Americans have acquired around 60 per cent of the Barclays Premier League club, while it is thought that the proposed deal values the club - who have spent the last five seasons in the top flight of English football - at around £110million.
The deal could see the expansion of the 21,000-capacity Liberty Stadium, the second smallest in the Premier League, as well as investment in the club's training facilities and playing staff.
Source : PA
Source: PA