Second-half goals from Juan Culio and half-time substitute Lacina Traore saw Cluj off to a flyer in their Europa League Group K opener.
The pressure will now ease slightly on Conceicao, whose future is the subject of speculation even at this early stage of the season.
Cluj had the best of the opening exchanges and Sebastian Dubarbier could have given them the lead in the 10th minute but he drilled his effort straight at Johan Wiland.
The Copenhagen goalkeeper was called into action again minutes later and he did well to save Cristian Panin's 20-yard thunderbolt.
The visitors were without influential winger Jesper Gronkjaer and it certainly showed as they struggled to create anything.
They did not have a sight on goal at all during the first half and could have found themselves several goals down at the interval were it not for Wiland and some woeful shooting by Cluj.
It was more of the same after the break and Cluj took the lead within 10 minutes of the restart when Sixto Peralta picked out Culio who blasted it home from 12 yards.
Traore should have doubled his side's lead on the hour mark but he sent his effort wide of the left upright.
The hosts were completely dominating the tie and it was only thanks to Wiland that Copenhagen were still in the match.
The Sweden goalkeeper pulled off another good save to tip Peralta's fierce shot around the post.
The hosts had their first real opportunity in the 70th minute but Cesar Santin headed wide of the right upright.
Moments later Peter Larsson also missed the chance to equalise as he drilled his attempt over the bar.
And they were to rue those missed opportunities as Cluj doubled their lead in the 75th minute when Traore fired into the top left corner from 12 yards.
Libor Sionko went the closest Copenhagen had come all night in the 83rd minute but Nuno Claro tipped his shot round the post with a fantastic diving save.
Cluj could have extended their advantage still further as Ciprian Deac rattled the crossbar as time ran out.