Arsene Wenger's men ended a nine-year trophy drought by claiming the FA Cup with an extra-time win over Hull City in May.
That success has given Arsenal confidence, and Oxlade-Chamberlain said the players were eager to build on the win.
"We all believed it would happen soon," he said.
"It was a reward for the hard work and togetherness we've had to show. It does start a winning momentum.
"Once you know you can win a competition like that, it gives you confidence.
"It's really exciting to see someone like Alexis Sanchez come in and our World Cup winners will come back and give us another lift. These are exciting times.
"It makes you want to strive for more. I've set myself the same goals we all would, which is to win the Premier League, the FA Cup and try to do really well and win the Champions League."
Oxlade-Chamberlain, who played no part in England's failed World Cup campaign, said his focus was on Sunday's FA Community Shield clash against Manchester City.
He praised Arsenal's signings – including the likes of Sanchez, Mathieu Debuchy, David Ospina and Calum Chambers – for adding competition for places.
"Obviously it's going to be tough for everyone to get in the team this season, and that can only be a good thing."