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."