The Gunners salvaged silverware from a season that failed to yield Champions League qualification, with Wenger coming in for sustained criticism amid a fifth-placed Premier League finish and a humiliating exit from Europe at the hands of Bayern Munich.
Goals from Alexis Sanchez and Aaron Ramsey ensured 10-man Chelsea were denied a domestic double in Saturday's Wembley showdown, as Arsenal secured a record 13th FA Cup – seven of which have come under Wenger's watch.
The veteran Frenchman has been evasive on questions over his future, but a decision is expected in the coming days and compatriot Coquelin wants to see him continue.
"I think it's a pleasure for everyone to offer this trophy to the coach," he said. "We will know in the next few days what he will do. Regardless of his decision, I think it was important for him to win this FA Cup. We are all proud to have done it.
"We hope he will stay, but it's not our decision, he is the only one who can take it. We'll see in the next few days. But we are happy to have won."
Wenger lashed out at his fiercest critics, citing a "lack of respect" that he described as a "disgrace".
Coquelin has no doubts about the 67-year-old's continuing influence at Emirates Stadium, which has made Wenger a key figure of Arsenal's identity.
He added: "I said it many times, when we hear the word 'Arsenal', the first name we think is Arsene Wenger.
"He did a lot of things for the club. I think fans should not forget it. Even though it has been a bit difficult in recent seasons. The fans want more. But I also know that the coach wants more and he will do everything to win trophies and we hope he will be there next season."