American Dudley, 64, will step down on February 4, 2020 and retire at the end March, the company said in a statement, adding he will be succeeded by upstream division boss Bernard Looney, from Ireland.
The announcement brings the curtain down on Dudley's 40-year career at the London-listed energy major, eventually steering it back into profit following the deadly oil spill tragedy that badly damaged its reputation.
BP clocked up vast losses on clean-up and compensation costs and was forced to axe thousands of jobs and sell billions of dollars of assets to pay a total bill of $70 billion.
Dudley was parachuted into BP in October 2010 after his predecessor, Briton Tony Hayward, was forced to quit amid heavy criticism of his handling of the disaster.
'Most challenging time'
"Bob... was appointed chief executive at probably the most challenging time in BP's history," said Chairman Helge Lund.
"During his tenure he has led the recovery from the Deepwater Horizon accident, re-built BP as a stronger, safer company and helped it re-earn its position as one of the leaders of the energy sector.
"This company -- and indeed the whole industry -- owes him a debt of gratitude."
The BP-leased Deepwater Horizon oil rig exploded off the coast of Louisiana while drilling a well 5,000 feet (1,500 meters) below sea level on April 20, 2010.
Eleven rig workers were killed in the blast that caused 134 million gallons (507 million litres) of oil to spew into Gulf waters.
It took 87 days to cap the out of control well, and the oil slick stretched across an area the size of Virgina, blackening beaches across five US states and hitting tourism and fishing.
The spill, one of the worst environmental disasters to strike the United States, prompted Dudley to improve its safety procedures and overhaul its structure.
"Sometimes events like this shake you to the core, the foundation, and you have two responses," Dudley had said in a media interview in 2010, in reference to the tragedy.
"One is to run away and hide, the other is to respond and really change the culture of the company and make sure all the checks and balances are there, just to make sure this does not happen again."
Meanwhile, under Dudley in 2018, BP bought mining giant BHP Billiton's US shale oil and gas operations in a landmark $10.5-billion deal that energised the company's output.
Push into renewables
In the latter years of Dudley's reign, BP increasingly moved into cleaner energies, mirroring the strategy of rivals like Royal Dutch Shell and Total, amid changing public attitudes and government policies towards carbon emissions and climate change.
BP snapped up Britain's largest electric vehicle charging firm Chargemaster in 2018, as it bets on booming demand for electric cars in the coming decades.
The company predicts that renewable sources could account for 15 percent of global energy consumption by 2040.
Looney, 49, joined BP in 1991 as a drilling engineer and rose through the ranks to lead its upstream division, which comprises exploration and production.
"Bernard has all the right qualities to lead us through this transformational era," Lund said.
"He has deep experience in the energy sector, has risen through the ranks of BP, and has consistently delivered strong safety, operational and financial performance."