Stephan Lichtsteiner had opened the scoring before Icardi nodded in and then set up the winner for Ivan Perisic.
“It hurts, because for 65 minutes we played badly and weren't very sharp in our passing, but we still went in front," Buffon told Sky Sport Italia.
“A side like ours cannot and must not lose from that situation. A draw can happen, but not a defeat. You take these results home with something other than tactics and fluidity. You need determination and grit.
“I think this defeat will do us good and it's perhaps normal not to be at 1,000 per cent hunger this early in the season. We were put under pressure on the pitch and need to improve things. This is the third goal of the season we've let in from a corner and that's really not like us.
“Aside from what the media says about our strength, which could well be true, we have to prove it on the pitch in every single game.
“After we beat Sassuolo 3-1 there was praise all round, but to be honest I had my doubts about that performance. We allowed Sassuolo four scoring opportunities and a side that is based on solidity can't be doing that.
“Against Sevilla we were criticised, yet I liked Juve for their solidity and ability to create chances. Today, aside from the beauty of our football, we could've won just through sheer solidity.
“I think one of Juve's strengths has always been our ability to concede one goal every two games, on average. If we score two per game, we generally win.
“However, we've conceded four in four Serie A rounds and that's not right for Juve."