Mikhail Fridman, representing the group of Russian billionaires that owns 50 per cent of the company, said the agreement "opens fundamentally new opportunities for the development of TNK-BP".
Neave Barker, Al Jazeera's correspondent in Moscow, said: "Now that a solution to the standoff has been agreed upon… it's up to TNK-BP to restore shattered investor confidence.
"Foreign investors in Russia have grown accustomed to the unexpected - but the accusations traded at TNK-BP hark back to the unstable business climate of the 1990s."
Dudley's departure was a key demand of the Russian shareholders who accused him of running the company purely in accordance with BP's interests and hindering its international expansion.
Many commentators have said they suspect investigations by state institutions such as the labour, tax and migration offices were used by the Russian shareholders to help push out Dudley and force other changes.
TNK-BP's British staff, including Dudley, left the country in July amid pressure from Russian authorities.
At the time, BP said Dudley had temporarily left Russia because of "sustained harassment" and would be running the company from outside the country.
Officials in the offices of Medvedev and Vladimir Putin, the country's prime minister, welcomed the resolution to the row.
Arkady Dvorkovich, Medvedev's economics advisor, said: "This resolution will benefit the shareholders, the company itself and naturally Russia as a whole.
"The resolution ... will undoubtedly have a favourable influence on Russia's investment climate."
Thursday's memorandum of understanding, due to be finalised in the coming months, was well received by investors, with BP shares trading around 3.6 per cent higher on the London Stock Exchange.
TNK-BP was formed in 2003, with 50 per cent owned by BP and the other 50 per cent by a group of Russian industrialists known as Alfa, Access/Renova.
BP's portion of TNK-BP accounts for a quarter of the British company's global production, 10 per cent of its earnings and a fifth of its reserves.
The company said Dudley would leave his post by the end of the year.
BP will nominate an independent chief executive for approval by TNK-BP's board, the two sides said.
The deal will also see a public flotation of up to 20 per cent of a TNK-BP subsidiary, BP said in a statement.
The row had aggravated relations between Britain and Russia which were already strained, notably by the 2006 death by poisoning in London of former Russian agent Alexander Litvinenko.
Russia's Vedomosti newspaper quoted one expert as saying the company was currently undervalued by 40 per cent on the markets as a result of the friction.