Roma's French defender Philippe Mexes smashed in the first after 69 minutes from just inside the area following a corner.
Substitute Marco Borriello then added a second when he scored with a superb trademark volley two minutes later.
The Romanians had come close in the first half when Lacina Traore hit the bar with a fierce shot after a strong run and then shook the post with another strike after the break.
Cluj kept battling away at a half-full Stadio Olimpico and deservedly pulled a goal back through Ionut Rada's header after 78 minutes but could not find an equaliser.
"I am satisfied," Roma coach Claudio Ranieri told Sky TV. "However, Cluj were very good on the counter-attack and their goal reopened a match which was in our hands."
The win puts Roma second in the group with three points behind Bayern Munich, who have six after a 2-1 win at Basel, with Cluj also on three and the Swiss side bottom with none.
"Today we did not play a very good match, apart from the positive result. We missed the brightness we had last season," Roma midfielder David Pizarro said.
Stand-in Roma goalkeeper Bogdan Lobont was almost embarrassed with a clearance that Cluj's Emil Dica attempted to send straight back past him but it flew narrowly wide.
The hosts did have other chances too with Francesco Totti's free kick well-saved by inspired Cluj goalkeeper Nuno Claro, who repeatedly kept Claudio Ranieri's side at bay. Mirko Vucinic also had a second-half free-kick palmed away.
Roma were boosted by their win 1-0 over treble winners Inter Milan in Serie A on Saturday which was their first victory this season, having lost at Bayern in their Champions League opener.
Cluj, who won at Roma on their Champions League debut two years ago, now go to Bayern on October 19 having beaten Roma's next visitors Basel in their first group game.
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