The American private equity giant has acquired majority control of OEG from Cyprus-registered Morcell, which bought the business in 2012.
Details of the transaction were not released. However, an industry source said KKR had invested around £100m in the deal.
It is likely to have allowed Morcell to book a significant profit on its investment. Morcell is owned by a London-based office that manages the affairs of wealthy investors.
Led by chief executive John Heiton, members of the management team at OEG have retained significant shareholdings in the company.
An industry source said the money invested by KKR included some expansion funding for OEG.
This will be used to help OEG to increase the size of the fleet of cargo and accommodation units that it manufactures and leases to oil and gas firms.They can be used to carry cargo safely to rigs and the like and to provide space for functions like laboratories.
It should also allow OEG to increase its presence in oil and gas markets around the world.
Dominic Murphy, head of KKR's operations in the UK, said: "OEG has built an impressive track record and has an exciting growth path ahead, with an eye toward expanding its fleet and geographic coverage."
Mr Heiton said: "We have achieved significant international growth to date, but our teams have the appetite and capacity to accelerate our expansion both organically and through acquisition."
He said KKR would help the company realise its global ambitions more quickly. OEG has regional hubs in the main offshore oil and gas regions, including Singapore, Australia and USA and facilities where it can load supplies in 20 other locations.
It has used acquisitions to accelerate growth during a period when surging activity in oil and gas markets around the world has boosted demand for the specialist units it provides.
Last month, the company moved into the Caspian area by acquiring Inverurie-based OSCA Environmental Services, for around £10m.
This gave OEG facilities in countries ranging from Russia to Angola.
In January, the company said it would invest a record £13m in 2014, expanding its rental fleet.
The deal with KKR is the latest in a series in which investors based outside Scotland have acquired oil and gas services busineses in the country.
Doughty Hanson led a £250m buyout of the Asco oil and gas logistics firm from London rival Phoenix Equity Partners in 2011.
America's Lime Rock Partners is backing the Enermech engineering firm in an acquisitive growth strategy.
Financial buyers expect favoured energy services firms to achieve rapid organic growth in coming years. Demand for oil and gas is expected to be underpinned by growth in areas like Asia.
The sector is fragmented, providing opportunities for financial buyers to support consolidation plays in which portfolio firms buy rivals. The enlarged businesses created can boost profitability by using their scale to get better terms from customers and suppliers and rationalising duplicated functions in areas like administration.
Private equity firms typically expect to exit investments after around five years by selling the businesses they helped develop.
OEG was acquired in 2008 by the Lonsdale Capital private equity business. It acquired the Vertec Engineering and Labtech rental operations in 2009.
In 2012, Mayfair-based Lonsdale said it had booked a big gain on its investment in OEG after selling the business to a private family office.
Last year, Lonsdale acquired a majority stake in OTEAC, an Aberdeen-based oil and gas assets integrity specialist.
The latest accounts filed by OEG at Companies House show it doubled turnover in the 18 month period to December 31, 2012, to £19.6m, from £9.84m in the preceding year.
It made a a pre-tax loss of £855,713 for the period, compared with £1.8m profit the year before.