His departure came a day after city rivals Universitatea Cluj ended their hopes of landing a fourth Romanian Cup trophy since 2008 with a 4-2 shootout victory in the semi-finals.
In January, former Romania midfielder Trica became the club's third coach this season.
They were second in the table at the time, nine points behind leaders Steaua Bucharest, but a month later Cluj were deducted 24 points because of their inability to deal with spiralling debts.
The Railwaymen owe nearly 20 million euros (14.61 million pounds) to the state, former players and coaches.
After two draws and four defeats under Trica, Cluj are bottom with 12 points from 23 matches, 13 adrift of safety.
Cluj won three domestic league titles between 2008-12.
(Reporting by Angel Krasimirov, editing by Tony Jimenez)
