Suchy saw red after just nine minutes on Thursday and when Jonathan Soriano put Salzburg 1-0 up on the night and in the tie, it was not long before the game had to be delayed due to Basel fans throwing missiles onto the pitch.
Play eventually resumed and two goals in 10 second-half minutes from Marco Streller and Gaston Sauro turned the game on its head before Alan was dismissed just before the end.
Salzburg flew out of the blocks and Alan headed over from a Florian Klein cross before Sadio Mane and Kevin Kampl both dragged efforts wide in the early stages.
Basel were on the back foot and their task was made harder in the ninth minute when Suchy was given his marching orders after a challenge from behind on Alan.
The sending-off seemed to spark the visitors into life and they were nearly ahead in the 19th minute as Valentin Stocker supplied a set piece from the right and Andre Ramalho did well to block Sauro's header.
But just three minutes later, they were behind with Soriano connecting with Kampl's low cross and slotting home from six yards.
Klein then brought an impressive save from Yann Sommer and the goalkeeper was called into action soon after as he deflected Mane's effort over the bar with his head just before the half hour.
Moments later, it all seemed to become too much for some of the Basel fans as they began throwing missiles onto the pitch causing a delay in proceedings, with referee Manuel Grafe taking the players back down the tunnel.
Play eventually resumed after a delay of roughly 15 minutes and Salzburg nearly had their second courtesy of the same duo that produced the first with Soriano dragging wide from Kampl's corner.
The stoppage seemed to take some of the sting out of the game and neither side could create another chance of note before the break.
It was the dissatisfaction of Basel fans that had caused the delay but they had something to cheer about six minutes after half time when their team grabbed a vital away goal.
Fabian Frei sent a corner to the back post and Streller powered home a header which the despairing Peter Gulacsi could only get his fingertips to.
Klein and Rodnei both missed the target with attempts to get the goal Salzburg now needed but the hosts were left with a major mountain to climb on the hour when Basel took the lead.
It was another corner from Frei that produced the goal and this time Arlind Ajeti headed back across goal after Gulacsi had missed with his punch and Sauro was on hand to head into an empty net.
Streller had the chance to put the tie beyond doubt shortly after but dragged his shot wide and Salzburg were clearly getting frustrated as they struggled to break down their opponents.
Alan had been lively all evening and nearly had the equaliser 10 minutes from time but Sommer was equal to his 18-yard drive and Ramalho could only head wide from the resulting corner.
Six minutes later, the Brazilian striker was dismissed for his second booking of the night and Salzburg's hopes of a comeback went with him.