Frederico Almerares scored the only goal to keep Basle's hopes of qualification from Champions League Group E alive.
Almerares netted 14 minutes into the first half against a Cluj side that rarely threatened the home goal at St Jakob Park.
In the first minute Basle midfielder Xherdan Shaqiri's long-range left-foot shot was off target.
Cluj's Juan Emanuel Culio should have done better after three minutes as he opted to chest down rather than head a cross but the ball dropped to goalkeeper Franco Costanzo.
Basle's Berang Safari got down the left wing but his deep cross was just ahead of Alex Frei and he was penalised for controlling the ball with his upper arm.
Fourteen minutes in, and a minute after being booked, Almerares got on the end of a low shot from Samuel Inkoom to open the scoring.
Inkoom had shaped to shoot from distance but scuffed his effort across goal and Almerares got behind his marker to shoot under Nuno Claro.
Within two minutes Basle could have been further ahead after Almerares worked space inside the area and passed to Frei but his shot was blocked by Ionut Rada and Felice Piccolo cleared off the line.
Almerares was gifted another chance on 20 minutes when a poor clearance fell to him but he fired over the bar.
Frei's corner was headed goalward by David Abraham but Rada got his head to the ball to concede another flag-kick.
Safari twice went close to doubling the lead but successive shots from inside the area on the half-hour were blocked by Piccolo and then Ricardo Cadu.
Rafael Bastos had Cluj's first real effort after nearly 35 minutes but following a strong run from his own half he shot well over from distance.
Culio won a free-kick when he was blocked by Inkoom but his set-piece was headed clear and Bastos shot over as the visitors opened the second half with more purpose.
Basle were denied just before the hour when a shot from Fwayo Tembo was deflected onto the bar by Nuno Claro's foot.
A minute later the Cluj keeper kept out Frei's shot with his foot and was then drawn out of his area but Tembo's cross was cleared.
Frei thought he had doubled the lead with 16 minutes to go but he was flagged offside on the edge of the six-yard area as he steered a shot from Tembo past Nuno Claro.
Cluj coach Sorin Cartu took out his frustration on the away side's dugout as he put his foot through the Perspex side.
Nuno Claro got down well to push aside a shot from Gilles Yapi Yapo as Basle continued in vain to press for a second goal.
Cluj played the last two minutes with 10 men after Piccolo was show a red card when he pushed over Almerares.