Michael Laudrup's men would have made the last 32 with a point after Kuban Krasnodar's 4-0 victory over St Gallen, but they must now wait until the final game in Group A to seal their progress.
Dani Parejo's strike wrapped up top spot for the visitors, who gained a measure of revenge for the 3-0 defeat they suffered in the reverse fixture.
The midfielder's first goal of the season earned Valencia its fourth consecutive win in the competition, with Swansea goalkeeper Gerhard Tremmel's poor clearance handing the Spaniards the game's only goal at the Liberty Stadium.
Only Michel Vorm, Alejandro Pozuelo, Wilfried Bony and Nathan Dyer remained from Swansea's last Europa League tie - a 1-1 draw with Kuban.
The returning Pablo Hernandez was named on the bench after returning from a hip injury but Michu remained absent due to an ankle problem.
Valencia made five changes from its 2-1 defeat against Elche on Sunday, with two of those coming in the back four as Joao Pereira and Victor Ruiz starting for Miroslav Djukic's side.
Bony almost put Swansea ahead when he headed just wide after Diego Alves had parried a corner to the Ivorian in a lively opening to the game.
Valencia, however, took advantage of Tremmel's clearance at the other end, as Juan Bernat's effort deflected into the path of Parejo who cleverly angled a finish home.
Jonas was next to threaten for the visitors, letting fly with a long-range effort, only for it to whistle over the crossbar, and Swansea was handed a further blow when Bony limped off injured.
The home side was further frustrated before the break, when Dyer looked to have leveled only for the linesman's flag to deny him, despite both he and Jonjo Shelvey, who headed the initial cross on target, appearing to be onside.
Sofiane Feghouli came close to doubling Valencia's lead four minutes after the break, shooting wide from a precise Sergio Canales through-ball.
Substitute Alvaro Vazquez looked to have equalized two minutes from time, only for Jeremy Mathieu to clear off the line as Valencia held on for the victory.
Follow GOAL.COM USA on Twitter
- Sports & Recreation