Dudley took over for Hayward last year after the fallout from BP's oil spill in the Gulf of Mexico.
Dudley told employees in an e-mail message in October and obtained by the BBC that fourth quarter pay bonuses were contingent on safety.
Dudley said the company was taking the step to be "absolutely clear" that safety was the top priority for the tarnished energy company.
"In particular, we are committed to ensure that a low-probability, high-impact incident such as the Deepwater Horizon tragedy never happens again," the message read.
BP is struggling to repair its reputation after the summer oil spill in the Gulf of Mexico. A gas explosion in April killed 11 workers and sank the Deepwater Horizon oil platform.
The accident led to the worst accidental offshore oil spill in the history of the industry, releasing nearly 5 million barrels of oil into the gulf.
BP last year offloaded some overseas assets to help pay for the estimated $11 billion in costs related to the spill.