A total of 200 jobs are to go from Lasmo's Bishopsgate headquarters in London - a quarter of the group's worldwide workforce - in a bid to save pounds 30m a year.
The two board members who are leaving, finance director Dick Smirnoff and John Hogan, who ran Lasmo's North Sea operations, are in line for pay-offs totalling around pounds 1m.
Both are on two-year contracts, although they are understood to be receiving 18 months' money. Mr Smirnoff earned pounds 298,000 last year and Mr Hogan pounds 286,000.
The rationalisation programme will cost Lasmo between pounds 30m and pounds 40m, to be taken as an exceptional charge in the current financial year, and will leave the group nursing a loss of around pounds 80m for the year. Analysts were already pencilling in a pounds 40m loss because of the collapse in oil prices.
The boardroom shake-up will result in a new management team.
Joe Darby continues as chief executive, but Chris Wright, the new business director, steps up to the job of group managing director. Paul Murray, corporate development director, becomes finance director.
Lasmo is also devolving its command structure to reflect its move away from traditional areas such as the North Sea into regions including Algeria, Libya, Pakistan and Venezuela where the group has major exploration acreage and reserves that can be exploited profitably even with oil prices at a 10-year low of $10 a barrel.
Under the new structure there will be six new business units covering Europe and North Africa, Indonesia, Venezuela, Libya, Pakistan and the Middle East. They will report to Mr Wright.
Mr Darby said the reorganisation was aimed at achieving a "radical and permanent improvement" in Lasmo's competitive position.
The cutbacks at Lasmo follow Royal Dutch Shell's move to cut 3,000 jobs in Europe - 20 per cent of its workforce - and close its London head office, Shell Mex House.