The Welsh lost 2-0 as the Viola through goals from Yacine Adli and Moise Kean.
Afterwards, Harrison said: "The big difference is that Fiorentina brought on three top players (Dodo, Kean, Albert Gudmundsson) and that changed the game. We tried to distribute our energy throughout the game, in the end I'm a bit disappointed about what happened in the second half, even though we did our best and in the first half we conceded two shots on target. There's also disappointment because the goals came from two dirty actions.
"I am enormously proud of where we are and to be able to represent Wales. Nobody ever likes to lose but this is a historic evening for the whole club. We have shown that we can play at this level. We faced a top-class team and we have asserted ourselves, we will see how it goes in the next five games."
On their plans in the ECL, Harrison said: "We came here without any experience at European level. We still have five matches and maybe by getting 7-8 points we can get back into the top twenty-four. However, I am still confident."