Following a goalless first leg, the night got off to the worst possible start for the visiting side, as Marek Suchy was shown a straight red card after a reckless challenge with just nine minutes on the clock.
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Soriano came close to adding a second with 32 minutes played, but he fired his effort wide of the target from a good position inside the opposition box.
The game had to be halted for 15 minutes in the first half due to objects being thrown onto the pitch from the away section, but play resumed with Salzburg creating further chances.
Despite being down to 10, Basel hit back in the second half courtesy of Marco Streller, who raised highest to nod home from a corner kick and put the visitors into an away-goals lead after 50 minutes.
Salzburg, who boasted a perfect home record in the Europa League ahead of tonight's clash, were reduced to 10 men with four minutes remaining to end any hopes of a comeback.