The ex-Gooner, who impressed en route to Chelsea's Premier League title triumph, was a second-half substitute against his former side as the 10-man Blues were beaten 2-1 at Wembley on Saturday.
"It wasn't easy to lose, but it was still a good season," he told reporters in the mixed zone afterwards. "We go home with a bad taste.
"But we would prefer to win the Premier League to the FA Cup, so it is still a good season."
While Fabregas took no issue with Conte picking Nemanja Matic in midfield, the Spaniard would not be drawn into a definitive answer on his long-term future amid summer transfer talk.
"It is between Conte and me [what he said]," Fabregas added. "I am not upset and I have nothing against Antonio. He has done a fantastic job.
"I think I have a very good relationship with him. He has been honest with me and me with him. I understand his idea of football very, very well. I like it a lot.
"Every player wants to play more. This is in the genes of a winner. I am one of them and I will never put my head down or hands down. I want to fight.
"[I'm] not necessarily [wanting to leave Chelsea], but, in football, you can never say never. Everything can happen.
"I feel loved by the fans here. Every time I have played, I have shown I can play at the top level in this team, by far. It is all about trying to change the manager's decision, but he wants what's best for the team."