PSG beat already-qualified Athletic Bilbao 4-2 at the Parc des Princes to hand the Spaniards their first loss of the tournament.
But the win was not enough as Salzburg won 3-2 at Slovan Bratislava to see the Austrians take second place in Group F.
That means there is no chance of the Parisians getting an attractive knockout round date with Champions League dropouts Manchester United, Beckham's former club, when Friday's draw is made at Nyon.
Also on the European scrapheap are English Premier League side Fulham, runners-up in 2010 but who squandered a two-goal lead to draw at home to Denmark's Odense.
The Danes bagged their leveller in the third minute of time added on at Craven Cottage through substitute Djiby Fall's header after Hans Henrik Andreasen had reduced the deficit following goals for the hosts from Clint Dempsey and Kerim Frei.
The result means Poland's Wisla Krakow progress instead after beating group winners FC Twente of the Netherlands.
Recent media reports in France had suggested that PSG's owners, Qatar Sport Investments (QSI), had offered Beckham a lucrative contract to become the figurehead of their drive to turn the under-achieving side into one of Europe's top clubs.
PSG, joint top of the French league table with Montpellier as they chase down a first domestic crown since 1994, spent 85 million euros in the summer following a Qatari buyout but the influx has not translated into success on the European stage and they now must ensure they reach the Champions League via a strong Ligue 1 finish.
Coach Antoine Kombouare has been under the microscope in recent weeks following some poor results and the pressure will now only increase with just the league title to aim for.
Kombouare was candid afterwards at his men's shortcomings.
"It's a big disappointment, we really should have got through this group. But we weren't able to do what we had to - now we have to get over it and do better in the league and French Cup."
PSG sporting director Leonardo denied that the axe would fall on the man in the dugout and told reporters: "That was one of our season targets. So we are disappointed. It cannot be good for us to be eliminated - we wanted to go right through to the end."
Elsewhere, Lazio reached the last 32 with a 2-0 win over Group D winners Sporting Lisbon.
English Premier league side Stoke went down 3-1 at Besiktas despite Ricardo Fuller giving them the lead but had already qualified from Group E.
Manuel Fernandes equalised with a penalty spot after Matthew Upson was sent off for felling Hugo Almeida, and substitutes Mustafa Pektemek and Edu secured the win at a heaving Inonu Stadium, leaving the Turks as group winners.
Eight of the 32 sides in the knockout round will comprise United and City's fellow Champions League failures -- Turkey's Trabzonspor, Ajax, Valencia, Olympiakos, Porto, and Plzen.
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