In a ballot run by the Central Arbitration Committee, staff voted 64% in favour of recognition and 36% against, representing a 53% yes vote across all 453 staff at the Greenwich-based museum.
The result overturns a decision by management in 1995 to derecognise unions and means NMM is no longer the only national museum not to allow unions to negotiate the terms and conditions of staff with management.
On behalf of Prospect members working as curators, conservators and technicians at NMM, Prospect negotiator Emily Boase said: "Our members are to be congratulated for overcoming years of resistance by management.
"Their determination will be rewarded as we can now seek to secure terms and conditions on a par with other national museums. Our research shows that since derecognition the amount spent on staff costs as a percentage of both grant-in-aid and income has reduced considerably."
Union figures show that in the years 1994-2000, direct grant-in-aid from central government increased by 27% and total income by 34.5%; yet spending on NMM salaries and pensions only increased by 14.2%. Cost-cutting changes to the pay and conditions of staff made during this period included abolishing the civil service pension scheme for new entrants and cuts in pay.
"Without an independent union to ensure that management invested in staff, funds were diverted to other projects, creating a gap in pay and conditions at NMM compared with other national museums," said Boase.
The NMM vote marks the second time Prospect has successfully used the legal framework introduced by the 1999 Employment Relations Act to establish rights to bargain collectively with employers.
Prospect represents 4,000 professionals across the heritage sector in national museums, galleries and libraries, including the British Museum, British Library, Tate Gallery, Natural History Museum, Science Museum and other bodies.