Trica, who won a domestic treble as a Cluj player in 2009, became the club's 13th coach since 2004 after replacing Portuguese Paulo Sergio who quit following the team's 3-2 home defeat to Pandurii Targu-Jiu on Friday.
Trica's appointment was confirmed in a short statement on Cluj's website, but there were no details on the length of his contract.
Sergio was named Cluj coach in October, replacing Ioan Andone, who had led the team to a domestic double in 2008 and to the title last season.
The Railwaymen, who have won three league titles and three national cups since 2008, are eighth in the standings with 36 points from 26 matches, 28 points behind leaders Steaua Bucharest.
They were knocked out of this season's Champions League at the group stage and were then eliminated from the Europa League by Inter Milan in the last 32.
Trica, 36, who also played for Steaua as well as teams in Bulgaria, Israel and Cyprus, will make his debut on Tuesday when Cluj host FC Astra in a Romanian Cup semi-final, first leg tie.
"I want to stay here three or four years," Trica, who has been coaching third-division side Juventus Bucahrest this season, told local media.
"Our first goal is to eliminate FC Astra in the Cup and bring the crowd, our 12th player, back to the stadium."
Former CSKA Sofia and Maccabi Tel Aviv player Trica is the youngest coach in the Romanian top flight.