Lenny Joseph scored twice in the second half to punish Milan, who struggled to contain the visitors despite playing at home. The forward opened the scoring after capitalising on a goalkeeping mistake by Lorenzo Torriani before doubling his tally 14 minutes later with a clinical finish on the counter-attack.
Milan struggled to respond and could not find a way back into the contest, even after Ferencvaros were reduced to 10 men when substitute Bamidele Yusuf received a straight red card in the 77th minute.
Amorim made no attempt to hide his frustration afterwards, insisting his players had failed to understand the responsibility that comes with representing one of Europe's biggest clubs.
"When you play for Milan, you have a responsibility; you can't go out there like this," the Portuguese coach said.
He also questioned whether his squad were mentally prepared to compete in both Serie A and Europe, insisting the problem had little to do with tactics or team selection. "We're not mentally ready. The technical aspect has nothing to do with it. The rotation has nothing to do with it," Amorim added.
The defeat leaves Milan 28th in the Europa League standings, with three losses from their opening four matches. That represents a major setback following their morale-boosting derby victory over Inter Milan at the weekend.
Amorim accepted responsibility for the performance but warned that his players must improve quickly. "I take full responsibility for this, because the team is my responsibility, but we can't afford to do something like this, especially in the first half, in front of our fans, at our home," he said.
Despite the setback, Amorim refused to give up on Milan's European campaign, promising that his team would continue fighting. "We're not ready to play every three days now, but I certainly won't give up on the Europa League. We'll fight until it's over. It's a promise."
Ferencvaros produced one of the biggest results of Europa League Matchday 4 with a 2-0 victory over Milan at San Siro, while Bournemouth and Celtic both staged impressive comebacks to claim three points.
Rasmus Hojlund has publicly disagreed with Massimiliano Allegri's assessment of Napoli's performance after their 4-1 Champions League defeat to FC Porto, insisting the Italian side must take greater responsibility for their poor form.
Como produced a remarkable performance to thrash Lens 9-1 away from home in the UEFA Champions League, equalling the biggest winning margin in the competition's history.
Parma stunned Atalanta with a convincing 3-0 victory in Bergamo to climb out of the Serie A relegation zone, while Fiorentina ended their five-match winless run with a thrilling 4-2 comeback victory over Sassuolo.
