The payout is about 70pc more than TNK-BP's first-half net profits of $2.43bn. Despite the huge shareholder dividend, TNK-BP has renewed its ambitions to expand through acquisitions and become an international player, according to its chief operating officer.
Bill Schrader, a former BP executive, said TNK-BP had the "total support" of the Russian government in its "big goal to internationalise the business".
The new strategy will be seen as a major victory for the Russian side of TNK-BP, which has long been pushing for a more global focus. In the past, it has been blocked by BP because of the oil giant's concerns that the two businesses could overlap and compete. The new strategy was agreed at a board meeting earlier in the summer.
TNK-BP is considering buying some of BP's assets in Vietnam, Venezuela and Algeria, but Mr Schrader said the process was going slowly.
"We're looking both at BP assets and other opportunities," he said.
BP and the four oligarchs are back on friendlier terms after a rift forced out TNK-BP's British chief executive, Bob Dudley, in 2008.
Mr Dudley took over the helm at BP at the beginning of this month.
TNK-BP declined to comment on the reason for the 8pc increase in its dividend.
The dividend will be useful to BP as it looks to fund the fallout from its Gulf of Mexico oil spill, with analysts estimating the total costs from the accident will top $30bn.