Dynamo Kiev were also knocked out after a squandering a two-goal lead and finishing with nine men in a 3-3 draw at home to Maccabi Tel Aviv as the evening's 12 ties produced 51 goals.
A last-minute goal cost 2010 runners-up Fulham their place in a 2-2 draw at home to Odense in more drama.
Lazio, Besiktas, former European champions Steaua Bucharest and Wisla Krakow reached the last 32, which will include Manchester United after their Champions League elimination last week.
Salzburg and PSG finished level with 10 points from six games in Group F, the Austrian side progressing thanks to their better head-to-head record.
Bilbao, led by eccentric former Argentina and Chile coach Marcelo Bielsa, had already qualified.
PSG, backed by Qatari investors, spent over 80 million euros on new players before the start of the season and their elimination is a setback for coach Antoine Kombouare after he appeared to have got back on track in Ligue 1.
Jon Aurtenetxe gave Athletic a third-minute lead but Javier Pastore, the most expensive of PSG's signings, equalised and Mathieu Bodmer put them 2-1 ahead by half-time.
"It's a great disappointment because to be honest, there was a good chance to qualify in from this group," said Kombouare. "I think we lacked realism during the group stage, especially against Salzburg and Bratislava."
Although David Lopez made it 2-2, an Inigo Perez own goal and a Guillaume Hoarau penalty in the last five minutes gave PSG the win which turned out to be in vain after Jakob Jantscher scored twice in Salzburg's win.
Salzburg, who have invested heavily since their takeover and subsequent re-branding by energy drinks manufacturer Red Bull in 2005, were 2-0 behind after six minutes after Milos Lacny scored twice for the Slovakian champions.
However, the Austrians, who have never reached the Champions League since the takeover, were level by the 25th minute thanks to a Jantscher penalty and a strike from Leonardo.
Jantscher then sidefooted the crucial goal in the 68th minute.
RED CARDS
Kiev, two winners of the old Cup Winners' Cup, had needed a large win combined with a Besiktas defeat to qualify from Group E and the evening began promisingly as Seran Yeini's own goal and an Oleh Gusev effort put them two ahead after 17 minutes.
However, they had Leandro Almeida sent off just after half-time and saw Omer Vered score from the resulting free kick before Yevhen Khacheridi was dismissed in an off-the-ball incident.
Elran Atar and Munas Dabbur made it 3-2 for the visitors but Gusev earned Dynamo a draw.
Besiktas came from behind to beat already-qualified Stoke City 3-1.
Fulham, needing a win to qualify from Group K, led 2-0 at half-time against
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