Rafael Santos Borre's goal put the semi-final tie beyond the Hammers and dumped them out of the competition after Aaron Cresswell was sent off for denying a goalscoring opportunity.
And as West Ham chased the game in vein, Moyes was sent down the tunnel for a furious outburst on the touchline.
The Scot appeared to boot the ball back at the German side's staff after trying desperately to retrieve it and continue the game, sparking a melee between the two benches.
Reacting on commentary, ex-West Ham striker Dean Ashton said: "He'll feel devastated at the way this game has gone. Inside, I bet he's boiling at what Cresswell did.
"That's his passion there. He wants his team to get going. He doesn't want time-wasting."
The home faithful created a white-hot atmosphere throughout the game and hundreds of fans invaded the pitch at full-time.
More to follow…