The Scot watched in horror as his defence was torn apart for Juraj Halenar to score a stunning hat-trick along with goals for Blazej Vascak and Martin Mikulic.
Demoralised and disbelieving, Celtic must produce a miraculous comeback at Celtic Park next week if they are progress to the third qualifying round of the competition - and not lose the minimum £10m jackpot on offer in the group stages.
Celtic are also likely to miss out on Uefa Cup qualification, since only the third-round Champions' League qualifying losers "parachute" into Europe's secondary competition.
Celtic's former defender Derek Whyte led the recriminations. "The financial implications are just unbelievable," he said.
"Players were getting caught out, they were disjointed, all over the place," Whyte said. "Their positional sense was awful at times. To lose five goals is incredible."
Strachan's men were already in shock after Chris Sutton was carried off on a stretcher early on with a suspected fractured cheekbone after a collision with his captain Neil Lennon.
Artmedia, the Slovakian champions, gave Celtic a warning in the 14th minute when Balazs Borbely made space for himself on the edge of the box before blasting his left-foot shot wide. Sutton, who started in midfield, hurt himself trying to make the block, so Strachan responded by bringing on the winger Aiden McGeady.
Bobo Baldé's incompetence almost conspired to put his side behind in the 23rd minute when he allowed Halenar to get behind him and take Daniel Tchur's chip into his path. But, with glory and Marshall in his sights, the Artmedia striker slashed wildly at the ball with his right foot and sent it hurtling well wide.
It was Halenar's first warning. Several Artmedia chances later, two minutes before the interval, Strachan had to tear up the half-time speech as Jan Kozak threaded the ball through to Vascak and he unselfishly squared the ball to Halenar to slot home.
And it went from bad to worse in the 57th minute when Halenar got behind the defence and squared for Vascak to score.
Maloney had two chances to pull Celtic back into the game, while McGeady then missed an open goal in the 67th minute. Eight minutes later, a corner broke to Halenar and he thundered a long-range volley past the despairing hand of Marshall.
Celtic's heads were down and the substitute Mikulic fired a fourth before Halenar rubbed salt into the wounds by completing a last-minute hat-trick to give Strachan plenty to ponder, such as Whyte's conclusion that "now he knows what it is like to be an Old Firm manager".
Artmedia Bratislava: Cobej, Debnar, Durica, Kotula, Vascak (Mikulic 72), Obzera, Tchur, Halenar, Borbely, Fodrek, Kozak. Substitutes not used: Kollar (gk), Stano, Hellebrand, Gomez, Gajdos, Burak.
Celtic: Marshall, Telfer, Varga, Baldé, Camara, Zurawski (Maloney 60), Lennon, Petrov, Thompson (Aliadière, 66), Hartson, Sutton (McGeady, 17). Substitutes not used: Boruc (gk), McManus, Wallace, Aliadière, Beattie.
Referee: Selcuk Dereli (Turkey)Reuse content