Ruthless finishing made the difference for Basle in Romania as they beat Otelul Galati 3-2.
And should they beat Manchester United on December 7 at St Jakob-Park, Basle will pip the English champions to a place in the Champions League knock-out stages.
That is the remarkable position which Heiko Vogel's team are in, after first-half goals from Fabian Frei, Alexander Frei and Marco Streller at the National Stadium in Bucharest kept up their challenge in Group C.
A late rally from Otelul Galati brought about two goals, but an equaliser was beyond the home side, who remain without a point after five matches of their first Champions League campaign.
Basle let their opponents see plenty of the ball in the early stages, yet by the quarter-hour mark and after only two attacks of note the visitors led 2-0.
The first goal arrived after 10 minutes when Alexander Frei took a few strides infield from the left flank and chipped a delightful pass to Benjamin Huggel inside the Otelul penalty area.
Huggel laid the ball off short to Fabian Frei and his firm and low first-time shot from the left embarrassed Branko Grahovac at the goalkeeper's near post.
It was 2-0 four minutes later when Streller raced into space on the left and, unchallenged, was allowed to zip into the penalty area.
He might have shot, as Grahovac came to meet him, but chose the smart option of chipping the ball across to Alexander Frei and the former Borussia Dortmund striker drove the ball home from 12 yards, one bounce and into the net.
Otelul, the champions of Romania last season, pushed for a reply but initially they lacked the clinical touch of the Basle forwards.
Gabriel Giurgiu was among the home players to spurn chances, firing over the crossbar.
Streller looked to have put the game safe for Basle in the 37th minute when he fired the third, running through the home defence after an attempted clearance from Cristian Sarghi went badly wrong. Streller flicked a smart shot with the outside of his left foot into the bottom right corner.
Basle pushed forward for much of the second half, seeking to increase their lead.
But it went to 3-1 in the 75th minute when Giurgiu crashed in a 20-yard strike, giving Basle goalkeeper Yann Sommer little chance as the ball found the top left corner.
And when the lead was cut to a single goal in the 81st minute, Basle were clinging on.
This time Liviu Antal was the scorer, capitalising on woeful play inside his own penalty area by Park Joo-ho and firing the ball past Sommer.
After four previous defeats, Otelul had a first Champions League point in their sights, but Dorinel Munteanu's side did not have a third goal in them.
Basle remain third in the group, but they stand just a point behind both United and Benfica, who shared four goals at Old Trafford.
Benfica are through to the last 16, on the basis of their superior record against United. Basle and United have everything to play for.