The Seagulls were thrashed 4-0 by the Giallorossi in the first-leg of their round 16 clash as Paulo Dybala, Romelu Lukaku, Gianluca Mancini and Bryan Cristante struck for the hosts.
De Zerbi later said: "We played our game. We gave our best. Maybe our best now is this performance.
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"I think Roma is much better in terms of experience to play this type of game and we suffer a lot. Too much I think.
"We are not used to playing this competition and today we paid everything. We paid (for) a squad not so long, we paid to play away in this stadium, we paid with too many injured players and for us, it is the first time so we have to improve.
"We have to progress and if we want to compete in this level, from the owner to the coach to the players we have to progress to reach one level higher."