Gonzalo Montiel scored the decisive penalty in the shootout for Sevilla after the scores had finished 1-1 at the end of extra-time. Paulo Dybala had put Roma ahead in the first-half before Sevilla drew level after the break via Gianluca Mancini's own goal.
Mendilibar said, "When they scored against us, you think that it will be very difficult to score against them. The equaliser came quickly in the second half and gave us the strength to continue in the fight to win the game.
"I'm happy to be here, to have won for the club, which was having a hard time when we got here. We have done something good that can have repercussions for the good of the club.
"When I take a holiday I will know what we have done. We started badly, they played comfortably. And then in the second half we changed and it was tough. In extra time nothing happened but we had the success of the penalties."