Los Che went into Wednesday's match at the Ramon Sanchez Pizjuan holding a 1-0 lead from last week's first leg, and looked in control after Roberto Soldado put them ahead in the 66th minute.
However, Sevilla midfielder Ivan Rakitic pounced on a mistake by Valencia keeper Diego Alves to pull one back for the hosts, before Victor Ruiz bundled the ball into his own net to level to scores on aggregate.
Nonetheless, Emery's men held on to progress on away goals, and had plenty of praise for his charges despite losing on the night.
"Our team had quality and maturity and we showed this by moving on in the tournament," Emery said in his post-match press conference.
"We could not handle Sevilla in the opening 20 to 25 minutes, when they put a lot of pressure and had good chances to score.
"From that moment on, we showed good personality and we had chances on goal. In the second half, we had control of the match until [Soldado's] goal.
"After that, we began to suffer but we managed to hold on."
Valencia will next meet local rivals Levante in the quarter-finals, after their neighbours saw off Alcorcon 5-2 on aggregate.