The stalemate sees the Yellow Submarine reach the final 2-1 on aggregate.
Emery said, "We are very proud because Arsenal are a very good team and we worked tonight, as well as in the first leg, and deserved to go through. We made a very serious match, my players constantly helping each other.
"We worked hard today, defended really well.
"Then sometimes we controlled parts of the match with possession. The best chances to score... I'd admit Arsenal had. I know it's going to be difficult against United but today we enjoy the moment. We deserve to play them and we will show our supporters our best performance."