The Gabonese striker is out of contract at the end of next season and has yet to confirm whether he intends to remain at the club beyond the summer.
Speaking to the BBC, Aubameyang praised his team's goalkeeper, Emiliano Martinez, saying: "Big respect for this man.
"We enjoyed today. We are so happy. As you can see, Emi [Martinez] is crying. That shows, that winning this game and this trophy makes us so happy.
"We deserved this win. He [Mikel Arteta] did a great job and we are so happy to have him on board."
Martinez said: "I can't really talk. I said to him [Pierre-Emerick Aubameyang] before the game, you are going to win us the game."
When asked about his future, Aubameyang said he could say: "Nothing. Just today and the trophy. That's it!"
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Arsenal manager Mikel Arteta was also drawn into the speculation of Aubameyang's future, saying: "He knows what I think about him. I want to build the squad around him. I think he wants to stay and it is just about getting the deal done.
"But I think these moments will help him to realise and believe that we are in the right path and he is a big part of that. He is loved by everybody at the club. Hopefully he can continue with us."
When asked if he thought the striker would sign a new deal, he said: "I think he will, yes."
Arsenal lift the FA Cup
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He also praised his players for their effort, proclaiming that part of the team's performance was the best football they had played under him as manager.
"It was a difficult start when we conceded so early. If I know something about these players, is that we weren't going to give up.
"We played probably the best 30 minutes after since I arrived. I am so proud to represent these players and this club.
"It is a double reward for us. It is really important for this club in every way to be in Europe. We all did it so I am so happy.
"The players were completely knackered and they made a massive effort through the coronavirus to keep fit so I am so happy for them."
César Azpilicueta (Chelsea)
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Conversely, Chelsea boss Frank Lampard was downbeat, and critical of his players after they failed to hold onto their lead.
"We started well for 10-15 minutes and we can only blame ourselves form that point. We got complacent, we took time on the ball like it was a stroll and we allowed them into the game. It's hard to get back into the game then. A lot of today is on us," he said.
"All I can do form the sidelines is shout! There are elements in our game that we have worked hard on all year but that is in you as a group. Today we were slow, we played back on ourselves, invited pressure. We didn't play well enough to win a final.
"I'm not even thinking about next week, it all came together for us today, two hamstrings, a dislocated shoulder. It's the end of a long long season and it was the tipping point for us."