Los Rojiblancos had claimed a 2-1 win in last week's first leg and they made light work of their opponents to romp to a 6-1 aggregate victory.
Suarez's miss mattered little as within minutes the hosts grabbed their second of the evening when Iago Aspas headed in Deulofeu's free kick.
Sevilla continued to dominate, but they had to wait until 10 minutes into the second half to score a third when Gameiro cut inside from the left and drilled a low strike beyond Roberto at his near post.
Gameiro then turned provider, as he picked out Suarez in the middle, who calmly slotted a low strike into the corner for his side's fourth of the evening.
With 11 minutes left to play Alfredo Ortuno almost got on the scoresheet for the visitors, but his powerful effort from range whistled inches wide of the post.
Minutes later Carlos Bacca threatened to add a fifth goal for the hosts, but after being slipped into the box his effort was stopped by Roberto.
In stoppage time Coke tested the keeper with a header that was pushed behind and it proved to be the last chance of the match as Sevilla confirmed a last-eight clash against Espanyol.