The EDF directors voted to accept a compensation package of 446 million euros on Tuesday.
But only just - six employee representatives voted against, six independent members voted for and government representatives abstained because of conflict of interest, according to sources.
In the case of a tie, managing director Jean-Bernard Lévy's vote decides, so the vote was in favour.
EDF is 85.6 percent state-owned.
The decision means that EDF should now ask the government to withdraw its permit to operate the reactor, opening the way for closure at the end of 2018, when a reactor in Flamanville, on the Channel coast, comes on stream, later than planned due to delays in construction.
The compensation package is much higher than the 80-100 million euros initially offered by Environment Minister Ségolène Royal.
2,000 jobs may go
The Fessenheim reactor's 850 employees have fought hard against closure, claiming that it is not a safety risk.
They cited a study that estimated the company will lose between 1.6 billion and six billion euros in income and appealed for it to be allowed to stay open for another 20 years.
More than 2,000 jobs will be lost, if the effect of subcontractors and suppliers is counted, according to another study.
Reliance on nuclear
Closing Fessenheim was one of President François Hollande's election campaign promises in 2012 and is enthusiastically supported by the Green party EELV, which joined his government but quit when Manuel Valls became prime minister.
While the contenders to be the Socialist Party's candidate in this year's presidential election and centrist Emmanuel Macron are in favour of closure, mainstream right candidate François Fillon has promised to keep Fessenheim open if he is elected.
A sustainable growth law passed last year aimed to reduce France's reliance on nuclear power by 25 percent from and the 75 percent share it currently has in France's power supply.