Details of the amount paid by the consortium and the total stake acquired have not been disclosed.
The takeover ends Italian businessman Francesco Becchetti's troubled three-year stay as O's owner.
The club recently staved off a winding-up petition at the High Court, having paid off debts to four creditors.
Following the dismissal of the winding-up petition, representatives of the Leyton Orient Fans' Trust (LOFT) had urged Becchetti to sell the club "urgently to any new responsible owner".
In a statement on the Orient website, Becchetti said the club had been sold without any "debts to the bank" and "without arrears for taxes".
"Over the last two years, I have, unfortunately, not been able to dedicate myself to following the club as closely as I would have wanted," he continued.
"This is clearly the moment to leave the club to others and I know that in doing so I am leaving it in good hands with Nigel Travis."
Travis, who describes himself as a "lifelong" O's fan, has been appointed as chairman of the club.
A second investor, Kent Teague, has been named as a director, while a fan representative will also be appointed to the board.
Local businessman Marshall Taylor has been appointed interim chief executive, while former Orient owner Barry Hearn will retain ownership of the club's stadium at Brisbane Road.
"Barry and I actually went to school together," Travis told BBC Radio London.
"Barry did a great job previously [as owner]. We're going to take some lessons from what he did and bring in some new stuff, but he remains the owner of the stadium and that's the way it's going to stay."
Travis has taken over the club in a personal capacity, with Dunkin' Brands, the parent company of Dunkin' Donuts and Baskin Robbins, not involved in the deal.
Leyton Orient had been beaten in the League One play-off final less than two months before Becchetti purchased the club in July 2015, but last season their 112-year stay in the English Football League came to an end with relegation from League Two.
The club had 11 managers or head coaches under Becchetti's ownership, including five different bosses last season alone.
They have been reported to have only 15 senior players under contract for next season, with the oldest of those players being 20-year-old Charlie Grainger.
Manager Omer Riza is only under contract until 30 June, with the club's first pre-season game scheduled for 12 July.
"We're going to make an announcement on the football side tomorrow [Friday]," said Travis.
"They will be the two key appointments. The idea is the board doesn't get involved in the day-to-day things."
A consortium led by Baskin Robbins and Dunkin' Donuts chief executive Nigel Travis has completed its takeover of Leyton Orient.