The Hammers went down 2-1 at home in the first leg before losing the reverse fixture 1-0 in Frankfurt on Thursday.
"Really disappointed," Moyes said. "Things didn't go for us on the night. Small things didn't happen but I am really proud of the players. How they played with 10 men was fantastic. They stayed in it. Other teams would have folded and lost two or three. The boys here are great. Tough boys.
"I thought we tried. We created chances. I thought we had better chances than Frankfurt. But we congratulate them. They go through, we don't and we give them our best wishes.
"I don't feel any of that [pride] at the moment. I just feel disappointment as I felt this was a chance. We have played better teams probably than Frankfurt. Let's be honest, we probably lost the game in the first 30 seconds at the London Stadium when we conceded. We were chasing the game since.
"We have enjoyed being in the competition. Not sure we have enjoyed the officiating. We have a pretty good team so I think we can try again. We will be a bit wiser and can hopefully do it more often."
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