The Serie A side needed only a point to finish second in the group behind Bayern Munich and did not waste much effort in a lacklustre performance at Cluj, converting only one of just three clear scoring chances.
Striker Marco Borriello beat the offside trap in the 21st minute, taking a neat through ball from veteran playmaker Francesco Totti into the penalty area and placing a low shot into the net.
Cluj striker Lacina Traore, who looked uninspired most of the game, equalised one minute before the end with a close-range header past goalkeeper Bogdan Lobont, a last minute substitute for Julio Sergio before the match.
Roma dominated the first 30 minutes but Cluj got back into the game at the end of the first half and Lobont had to make a brave save against a close-range volley by striker Roberto De Zerbi.
Left-winger Juan Culio then sent a powerful shot inches over the bar 15 minutes into the second half, while Totti and Borriello missed their shots from tight angles in the last half hour.
It was the first Champions League game for caretaker Cluj coach Alin Minteuan, who replaced Sorin Cartu after he smashed up the dugout during the 1-0 away defeat to Basel.
The match was low key, with Cluj already out of the competition and Roma confident at half time that a clear advantage for group leaders Bayern Munich at home against Basel would see them through without effort.
