Danilo D'Ambrosio and Lautaro Martinez struck the goals in each half, although Napoli kept possession for long periods at San Siro.
“You have to put the ball in the net. If you have five or six chances and don't put it in the net, of course you're going to struggle," Gattuso told Sky Sport Italia.
“I thought that of the four games against Inter this season, between Serie A and the Coppa Italia, this was our best performance. We created numerous scoring opportunities, had a different approach to the game, but we didn't make it count.
“Barcelona are a completely different team, as Inter are very physical and aggressive, whereas Barca pass it around much better. We've got to work on ourselves, because we are creating the chances, but we lack soul.
“We conceded two goals that were avoidable and didn't finish off the chances we created, that comes from a lack of soul. We had it, we've lost it lately and need to get it back.
“Quality only gets you so far. It's not like you can switch your soul on and off in a few days, we've got to get it back as soon as possible. This isn't enough. We hit rock bottom this season, we got it back on track and after the Coppa Italia we've been putting in good performances, but not with the bit between our teeth.
“The tempo slows when we get sluggish, when we start acting like traffic wardens going pass it right, now pass it left. That's no good. Anything I tell you on television about my team, I have told them first. I hide nothing and they know what I think.
“We weren't there in the opening 10 minutes, we were low on intensity and cannot afford that against Inter. After that, I liked the team, we stayed in their half pretty much constantly, but we didn't finish our chances. Arek Milik did not have a good game, I expected more from him tonight."