The hosts raced to a two-goal lead after only 12 minutes in the battle of the group outsiders and the Swiss never recovered from the early blow despite dominating much of the rest of the game.
Taking part in their third Champions League campaign, FC Basel had won all four qualifying matches but could pay dearly for failing to take home at least a point in a group which also includes last season's runners-up Bayern Munich and AS Roma.
"I'm very disappointed and so is the team, it's tough," said midfielder Benjamin Huggel. "We had counted on a point from this game."
Cluj, in their second Champions League campaign, made the astonishing decision on Monday to fire Andrea Mandorlini after an indifferent start to the season, even though he led them to the double last season.
Sorin Cartu took his place and could not have got off to a better start as his new charges raced ahead with two goals set up by Argentine midfielder Juan Culio.
Defender Ionut Rada rose above the Basel defence to head in Culio's corner after nine minutes, then lanky Ivorian striker Lacina Traore headed the second after Culio chipped the ball back from the byline following a quick counterattack.
Cluj then dropped back into defence and the game deteriorated. With dozens of empty seats and no stand at all behind one of the goals, it felt anything but a match in club football's richest and most prestigious tournament.
Cluj goalkeeper Nuno Claro made his first save just after the half hour, turning a Huggel header over his crossbar, but was beaten in first-half stoppage time when Stocker scored with a near post header from a corner.
Basel, who also won the double last season, had plenty of possession in the second half although they failed to create many clear-cut chances while Traore was a constant handful for the Swiss defence at the other end as Cluj played on the break.
Substitute Scott Chipperfield had Basel's best effort when his header was tipped over by Claro while Switzerland's record scorer Alexander Frei should have done better with a 20-metre free kick which he fired straight into the wall.
"I'm happy with the performance but not with the result," said Basel coach Thorsten Fink.
