Federico Almerares, one of five Argentines on the pitch who are all relatively anonymous in their homeland, scored the winner in the 15th minute as Basel made off with the points despite arguably their worst home performance in the group.
Basel were left with six points from five games, three points behind AS Roma who came from behind to beat group leaders Bayern Munich 3-2.
The Swiss need to win at Bayern Munich in their final game and hope the Italians lose at Cluj, who are already eliminated.
Neither result is likely as the Romanian champions have lost four games in a row in the group since beating Basel 2-1 in their opener.
Basel were the last Swiss team to reach the knockout stages back in 2002/03. Since then, FC Thun have been knocked out in the group stage in 2005-06, Basel in 2008-09 and FC Zurich last season. In the other intervening seasons, they did not have a team in the group stage.
SATISFIED COACH
"It's a shame Bayern Munich lost but we are still going to there to win, we have nothing to lose," said Basel coach Thorsten Fink. "I'm very, very satisfied with the performance, a coach is always happy when his team keeps a clean sheet."
Almerares, who made only a handful of appearances in three seasons at River Plate in his homeland before joining Basel two years ago, was standing in for the injured Marco Streller.
The 25-year-old stabbed home the winner from an unmarked position after Samuel Inkoom's shot found its way through a crowd of players.
Basel completely controlled the game but, as before in their campaign, they failed to convert their dominance into goals.
Behrang Safari saw his shot cleared off the line by Cadu, then Alexander Frei, the record scorer for Switzerland at international level, missed his kick when Fwayo Tembo's cross had left him in the clear.
It was a similar story after the break as Tembo saw his shot deflected on to the underside of the bar by Nuno Claro and Frei's effort from the rebound was also blocked by the Cluj goalkeeper.
In a scrappy and niggly second half, Frei had a goal harshly disallowed for offside after turning Tembo's pass into the net from close range.
Cluj, guaranteed to finish bottom of the group, finished with 10 men after defender Felice Piccolo was given a straight red card near the end.
