Paulo Sergio, 44, is Cluj's 12th coach since 2004 and the second Portuguese to take the reins this year after Jorge Costa left the club in April.
He has had spells at Vitoria Guimaraes and Sporting Lisbon and joins Cluj, who have won three league titles since 2008, after leading Hearts to the Scottish Cup in May.
"I am a modest guy," Paulo Sergio told a news conference. "I like to be discreet and work. Success comes after work."
The club are currently second in Champions League Group H, having won one, drawn one and lost only to leaders Manchester United in their opening three matches.
They have struggled domestically, however, and are currently seventh in the league with 19 points from 13 matches, 13 points behind leaders Steaua Bucharest.
Paulo Sergio, who will make his debut in the dugout on Wednesday when Cluj host second division club Botosani in the Romanian Cup, will have to win over fans who criticised Andone for his over-defensive coaching style.
"Ranking doesn't look good, but we don't have to think about rankings now," said Paulo Sergio.
"Nobody wants to win more than me and I know that step by step we can achieve our goals."