The Gunners ended a 10-year trophy drought in May when they came from two goals down to beat Hull City 3-2 at Wembley.
Arsenal boss Arsene Wenger has already added to his squad by signing Chile international Alexis Sanchez from Barcelona for a fee of around £30million.
Reports in Spain suggest Arsenal are also intent on bringing in Sami Khedira from Real Madrid, while compatriot Lars Bender, who missed the World Cup through injury, is another reported target.
And Jenkinson reckons Arsenal will be well equipped to challenge for other trophies, especially with the FA Cup win still fresh in the mind of players already at the club.
Jenkinson said: "Once you have got one, you want to experience it again, so fingers crossed we can compete on all fronts this season.
"We have the squad and we are signing players like Sanchez, so I don't see why we can't.
"He is world class and they are the sort of players you want to see at the club.
“The squad are buzzing as much as the fans, because if you are going to push on as a club and really compete for things, you need to sign top players.
"We have done that with him for sure - it is a fantastic addition.
"He's quick and direct, he can play up front and on the wing, and he is very dynamic. He will definitely fit the Arsenal style of play."