LONDON—BP PLC's chairman backed embattled Chief Executive Bob Dudley Thursday as shareholders at the company's annual meeting slammed the CEO's handling of BP's proposed Arctic-exploration deal with Russia's state oil company. They also questioned the U.K. oil giant's safety record in the wake of the Deepwater Horizon catastrophe.

Meanwhile, Mr. Dudley said BP, in a bid to smooth its proposed alliance with Russian state oil company OAO Rosneft, had offered to buy out its billionaire partners in its...