Highly-rated winger Dennis Man scored the winner from close range after a defensive mix-up midway through the second half to earn FCSB their first trophy under their new name.
FCSB, renamed after a court confirmed that the defence ministry had the rights to use the Steaua brand and symbols related to it, are fifth in the league standings, 16 points behind leaders Universitatea Craiova.
Steaua, the 1986 European Cup winners, were founded in 1947 as an army club.
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Sepsi, the first team from the ethnic Hungarian-majority county Covasna to win promotion to the top flight, finished the game with 10 men after Algerian defender Rachid Bouhenna was sent off 10 minutes from time.
Sfantu Gheorghe-based Sepsi, founded in 2011, are receiving financial backing from the Hungarian government which has sparked controversy in Romania.
(Reporting by Angel Krasimirov, editing by Ed Osmond)
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