The former Soviet Union striker took over from Daniel Isaila in October but led the side to just two wins in nine league matches.
Astra are fifth in the table with 32 points from 19 matches, 12 points behind leaders Steaua Bucharest.
Protasov's team were also beaten 3-0 by Steaua in the first leg of a League Cup semi-final last month.
The much-travelled 51-year-old previously coached clubs in Greece, Cyprus, Russia, Kazakhstan, Belarus and his native Ukraine.
During his playing career Protasov scored 29 goals in 68 matches for the Soviet Union between 1984-91. He was also capped once by Ukraine.