The company said investors holding 75.7 percent of the shares voted were in favor of the deal in a preliminary count.
"We, from a management perspective, from a board perspective, truly believe this is the right transaction for us," Bristol-Myers Chief Executive Giovanni Caforio told reporters after the vote.
"The focus is on us right now to execute on the integration and then deliver the value of the combined portfolio, to confirm that the new company will deliver significant value for shareholders," he said.
Celgene said separately in a statement that its shareholders representing more than 70 percent of its shares outstanding who were entitled to vote, voted in favor of the transaction.
The companies expect the deal to close in the third quarter.
Bristol-Myers announced in early January that it planned to buy Celgene in a cash and stock transaction to bring together companies that specialize in oncology and cardiovascular drugs in what would be the largest pharmaceutical industry merger ever.
The New York-based drugmaker has said the combined company will have six drugs with expected near-term launches - five from the Celgene pipeline - representing over $15 billion in annual revenue potential - as well as strong early-stage experimental assets.
But Starboard and the company's second largest shareholder, Wellington Management, opposed the deal.
Starboard called it "poorly conceived and ill-advised," and criticized Bristol-Myers' management and board, suggesting they would not be able to successfully execute a risky deal.
Starboard abandoned the campaign against the deal late last month after proxy advisory firms recommended investors support it.
Bristol-Myers said during a shareholders meeting that it does not expect changes to its dividend policy after the acquisition.
Bristol-Myers shares fell 50 cents, or 1.1 percent, to $45.59 in Friday afternoon trading on the New York Stock Exchange. Celgene shares fell 9 cents to $94.14 on the Nasdaq.
(Reporting by Michael Erman and Svea Herbst-Bayliss, writing by Caroline Humer; Editing by Chizu Nomiyama and Bill Berkrot)
By Michael Erman and Svea Herbst-Bayliss