Morata scored a brace as Juve brushed Ferencvaros aside 4-1 in the Champions League on Wednesday.
The forward has scored six goals in his past five away Champions League appearances, one more than in his first 20 games on the road in the competition.
Morata, who has six goals in seven games this season, said he was happy in his second stint in Turin.
"We are happy with the three points obtained, it is important to have won two games after a period of adaptation, but we know we can still improve a lot," he said, via the club's website.
"If we continue like this we can go far. I feel good, more complete, I am no longer 20 and what I have experienced in my career has helped me to grow.
"I have everything I need, I feel appreciated and I have great companions."
Juve scored three times in the final half-hour in Hungary, with a Paulo Dybala strike and a Lasha Dvali own goal after Morata's double seeing them 4-0 up before Franck Boli denied them a clean sheet.
The Serie A giants have won 10 of their past 14 away Champions League group stage games, including four in a row.
Juve defender Giorgio Chiellini urged his side to continue improving after moving onto six points in Group G.
"We are satisfied with the result, but we could have done better as a game," he said.
"We made a few too many mistakes, also due to the fact that the match was going downhill, but we need to improve.
"We have to make the right choices and be less selfish, close games earlier, be more aggressive and follow the insertions with the midfielders."
Sitting third in the league, Juventus are away to Lazio on Sunday.