Spanish striker Cristian Lopez converted a penalty in the 41st minute, his fifth league goal this season, and substitute Claudio Vioculet made it 2-0 with a thunderous long-range shot 13 minutes from time.
Dinamo's Cameroon right back Patrice Feussi was dismissed for collecting a second booking five minutes before the break and left back Steliano Filip was also sent off for two yellows nine minutes after the interval.
The 18-times Romanian champions were then reduced to eight men two minutes from time when midfielder Cosmin Matei was dismissed for spitting.
Dinamo, who won their last league title in 2007, are fourth in the standings with 17 points from 10 matches, three points behind leaders FC Astra.
Cluj, who began the season with minus six points after failing to put their financial affairs in order and meet licensing criteria, are seventh on 12 after their first away win of the season.
(Reporting by Angel Krasimirov, editing by Alan Baldwin)
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