Southampton face a final matchday showdown with Hapoel Be'er Sheva to ensure their Europa League progression, after they slipped to a 1-0 defeat to Sparta Prague on Thursday.
Costa Nhamoinesu's 11th-minute winner secured a narrow victory for the Czech hosts, sealing their place in the knockout stages with a game to spare.
Southampton, who have now scored just five goals in their last eight games in all competitions, face Hapoel Be'er Sheva in their final Group K fixture - at St Mary's on December 8 - and must win or draw 0-0 to advance.
Claude Puel made seven changes from the goalless draw against Liverpool, and his side had the first opportunity of the game.
Jordy Clasie picked out Jay Rodriguez but the Saints forward dallied in possession, allowing Sparta to scramble clear.
The hosts were far more clinical and went ahead when poor Saints marking allowed Nhamoinesu to convert Borek Dockal's free-kick.
Southampton responded and dominated the remainder of the first half, with Sam McQueen positive down their left.
He forced a smart save from goalkeeper Tomas Koubek before setting up Rodriguez, only for his shot to be well blocked from 12 yards.
In the second half, Pierre-Emile Hojbjerg picked up a dead leg and was replaced by Oriol Romeu, while Puel also threw on Sofiane Boufal and Charlie Austin.
But Sparta created the better opportunities after the break, with Fraser Forster called into action to deny Dockal twice.
Lukas Julis then clipped the top of the Southampton goalkeeper's crossbar, while Lukas Marecek fired narrowly wide as the home side pushed for a second.
Virgil van Dijk squandered Southampton's best chance of an equaliser, powering a header over from close range.
And in the end it was comfortable for the Czech outfit, who held on to earn their place in the last 32.
Southampton face the prospect of a nervy home encounter with Hapoel Be'er Sheva, who overcame a two-goal deficit to beat Inter Milan on Thursday.
If the Saints win or secure a goalless draw against the Israeli opposition, they will qualify second behind Sparta.
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