Both Chiellini and Rossi were on the same track, but when that cross was kicked in, Chiellini was still running to the middle while Rossi made a move to the far wide. You can see that in the replay, the moment that both Rossi changed direction and the cross was kicked were almost the same (miliseconds difference at least). (My guess is) Chiellini thought that the cross was going to be a little low because he saw Rossi made cut in runs the last time Rossi was in his view.
Chiellini had to focus more on the crosser because he knew that cross will go in anytime at that moment. He didn't have time to check Rossi's movement, well if he did, he'll lose the track of the ball and everything will be too late, not that it matters anywyay. He made judgment calls and unfortunately it didn't go his way.
What matters the most is the cross; that cross was precise even though Rossi changed direction almost at the same time the cross was made.
In the end of course I think Chiellini should do better but it's barely a fuck-up. This is not a video game where you can track the attackers runs that easily; Rossi played in Chiello's blind spot nicely and the cross was really good.