Liverpool's hopes of reaching the last 16 suffered a blow after they were narrowly beaten by Monaco.
Monaco took the lead early in the second-half after Javier Saviola struck home a low shot, though he appeared to handle just before shooting.
Steven Gerrard twice tested Monaco goalkeeper Flavio Roma, who also saved well from a Sami Hyypia header late on.
But with striker Neil Mellor playing his first European game, Liverpool were badly short of ideas in attack.
Saviola was Monaco's most dangerous player, though in the opening half he had been well shackled by Liverpool.
Monaco struggled to find any rhythm to their passing for long periods of the game, but the move leading up to the goal, involving Jaroslav Plasil and Shabani Nonda, showed the Ligue 1 side at their best.
Even so Traore's tackle on Nonda looked to taken the sting out of the Monaco attack, but the ball broke to Saviola, who whipped a shot past Chris Kirkland.
As Liverpool searched for an equaliser Kirkland was pressed into action more and more and he made a quick reaction save from Pontus Farnerud and then rushed out to deny Emmanuel Adebayor.
Deprived of Milan Baros, Liverpool manager Rafael Benitez had left Xabi Alonso and Harry Kewell on the bench, opting to play Mellor up front as a lone striker.
Luis Garcia played wide on the right but the former Barcelona player pulled up after only three minutes and was stretchered off with a hamstring injury, to be replaced by Josemi.
Just past the hour Josemi himself had to go off after he was involved in a bone juddering clash of heads with Patrice Evra.
On a bobbly surface Liverpool initially tried to isolate the Monaco left-back Douglas Maicon, who was put under pressure at every opportunity by John Arne Riise.
On the same flank Saviola looked to do the same to Traore.
Twice Saviola looked to have got away from Traore, but the Liverpool defender recovered superbly.
After a corner Saviola did manage to get a shot in, which Jamie Carragher showed good judgment in letting the ball run past him and out of play.
Carragher also made two important interceptions to take the ball away from Nonda as Monaco threatened just before the interval.
Liverpool's best chance in that scrappy first-half fell to Igor Biscan from a long throw, but the Croatian sliced a difficult bouncing ball wide.
With Olympiakos defeating Deportivo la Coruna, Liverpool must beat the Greek side when they meet at Anfield on 8 December to keep alive their chances of making the next round.
Monaco: Roma, Evra, Plasil, Maicon, Zikos, Nonda, Squillaci,
Farnerud (Modesto 79), Givet, Chevanton (Perez 67),
Saviola (Adebayor 73).
Subs Not Used: Audard, Kallon, Oshadogan, El Fakiri.
Goals: Saviola 54.
Liverpool: Kirkland, Finnan, Carragher, Hyypia,
Traore (Kewell 58), Gerrard, Hamann, Biscan, Riise,
Luis Garcia (Josemi 4), Mellor, Josemi (Warnock 64).
Subs Not Used: Dudek, Alonso, Diao, Sinama Pongolle.
Booked: Traore, Hamann, Warnock.
Ref: Claus Bo Larsen (Denmark).