The Gunners are through to the fifth round of the competition after defeating Leeds United 2-0 at home on Tuesday.
Eddie Nketiah and Chambers got the goals for the Gunners to win the game, with Chambers scoring within seconds of coming off the bench.
"We want to go all the way [now]," Chambers said.
"A club like Arsenal needs to be going all the way and competing for silverware in every competition, so that's obviously our aim.
"That's our goal, we're fully focused on that and believe we can do that."
"All the guys on the bench who were putting me on, the set-piece coach Nico [Jover], the boss, they all said: 'Go and score on your first touch'. So that was all that was in my head," he added.
"It was nice to see it come off my head and go over the line. It was definitely a first and a pretty unique way to score."
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