It was the Tynecastle side who took the lead when Dennis Wyness grabbed his first European goal but they thought they would have to settle for a point when Cesar Andres Carignano grabbed a late equaliser.

But the match took another twist when Neilson grabbed the winner with just one minute remaining.

Hearts headed into the game knowing they had to leave the Stadion St Jakob-Park with at least a point if they wanted to keep their European dream alive following a disappointing start to the group stages.

The opening two games against Feyenoord and Schalke left them without a point and they had the added disruption of changing managers midway through the UEFA Cup campaign.

John Robertson’s first game in charge was the defeat to Schalke but he emerged from that game insisting Hearts could win their last two games.

But it was Basle who looked the stronger side in the early stages and they came close when a Scott Chipperfield cross found the head of Benjamin Huggel. He nodded just wide after just three minutes.

Only the goalkeeping skills of Craig Gordon prevented Hearts from falling behind in the 18th minute.

Matias Emilio Delgado tried his luck with a thunderous long range effort that Gordon did well to block.

Julio Hernan Rossi pounced on the rebound and tried to send his header past the Hearts goalkeeper from six yards but Gordon was again on hand to deny the home side.

Hearts had a great chance to take the lead when Michael Stewart played the ball through to Wyness, who took one touch with his right foot before shooting with his left but dragged his effort across the face of goal.

But Hearts did not have too much longer to wait for the opener as they took the lead after 31 minutes.

Ramon Pereira started the move when his pass found Hamill who set up Wyness for the shot from just outside the six-yard box and he left the Basle goalkeeper with no chance when he fired home.

Basle resorted to desperate measures to haul themselves level again when Mile Sterjovski claimed he had been hauled down in the box by Alan Maybury but referee Kristinn Jakobsson ignored the claims.

Gordon then pulled off another superb save to prevent Basle from levelling the scores.

Sterjovski’s cross found Chipperfield inside the six-yard box and his diving header looked certain to find the back of the net until Gordon sprang into action to deny the Australian.

Basle were unfortunate not to find the back of the net when Chipperfield’s corner picked out Carignano and he watched in agony when Maybury hooked his header off the line.

But Carignano was celebrating after 76 minutes when he managed to shrug off Maybury before firing home from eight yards.

Just when Hearts thought they would be forced to settle for a point, they left Basle reeling by taking the lead again with a minute to go.

Phil Stamp’s header fell to the feet of Neilson and he left the Basle goalkeeper with no chance with a thunderous strike.