(Artmedia win 5-4 on aggregate)
Brave Celtic fell just short of creating European football history as they exited the Champions League despite a four-goal win over Artmedia.
They had to become the first side to ever progress after a 5-0 first-leg defeat but went out on aggregate.
Maciej Zurawski won a penalty and Alan Thompson fired in the opener.
John Hartson drove in the second, a Stephen McManus header deflected in for the third and substitute Craig Beattie nodded in the fourth.
Celtic boss Gordon Strachan, minus the injured Stan Varga and Chris Sutton, had made several changes as he attempted to make up for the humiliating 5-0 defeat in Bratislava.
Wallace, making his first start of the season, was set up by Hartson for an early chance, but the winger fired a couple of feet over the bar from 16 yards.
Zurawski sent a low drive across the face of the Artmedia goal and the Pole was then brought down inside the box by Ondrej Debnar after a knock down by Hartson.
Thompson stepped up to convert the spot kick to give Celtic 70 minutes to score another four goals.
Artmedia, despite bringing in defender Burak for Kotula, did not appear ultra-defensive in an attempt to protect their lead.
And Artur Boruc, making his competitive Celtic debut in place of David Marshall, saved well at the feet of Balazs Borbely, while Juraj Halenar drove across the face of goal, as the home defence remained vulnerable to the break early on.
Juraj Cobej did well in the other goal to push away a rasping free-kick from Thompson.
Hartson increased Celtic's lead on the stroke of half-time as he pounced on a Wallace header, turned his marker on the edge of the box and fired the ball into the far corner.
Substitute Shaun Maloney, on a matter of seconds, set up the third after 54 minutes, driving the ball into the six-yard box, where McManus' header deflected past Cobej.
Hartson and Maloney both fired over, while Cobej saved well from the Celtic substitute and fellow replacement Craig Beattie.
A Stilian Petrov overhead kick slipped just wide as the Slovakian champions lived dangerously in a second half during which they rarely threatened at the other end.
Substitute striker Beattie headed home the fourth goal, but Celtic were left to ponder what might have been had they not conceded so many in Bratislava.
Celtic: Boruc, Telfer, Balde, McManus, Camara, Petrov, Lennon, Thompson (Aliadiere 87), Wallace (Maloney 53), Zurawski (Beattie 63), Hartson.
Subs Not Used: Marshall, McGeady, O'Dea, Lawson.
Booked: Thompson, Balde.
Goals: Thompson 22 pen, Hartson 45, McManus 54, Beattie 82.
Artmedia Bratislava: Cobej, Obzera, Debnar, Durica, Vascak (Mikulic 74), Tchur, Halenar (Bukvic 87), Borbely, Fodrek, Burak (Stano 67), Kozak.
Subs Not Used: Hellebrand, Kotula, Gomez, Kollar.
Ref: Alberto Undiano Mallenco (Spain).