The Spaniard, speaking at the PUMA kit launch, believes the Gunners can build on their FA Cup win in 2014-15 if they can keep key players like Mesut Ozil and Aaron Ramsey fit.
Arsene Wenger’s side ended its nine-year run without a trophy by beating Hull at Wembley in May and has made a splash in the summer transfer window by signing Alexis Sanchez.
With the likes of Mathieu Debuchy and Sami Khedira also linked with a move to north London, Cazorla is confident Arsenal can hit new heights in 2014-15.
“We hope to win more trophies - I want to win the Premier League or Champions League,” he said at the launch of the club’s new PUMA kits. “Of course we can win. We have a good team and with the support of the fans it's important for us.
“We have high expectations to go better than last year. We won a trophy last year and we want to do it again. Hopefully we'll make some great signings to support the squad and if so, it'll be good.”
The Spaniard hopes the Gunners’ chances will be boosted by the returns to fitness of Mesut Ozil and Aaron Ramsey, who both missed spells of last season due to injury.
“I'm very happy to have Ozil next to me,” Cazorla said. “We have a very special relationship. He makes it very easy for me. Obviously he'll help the team to improve next season. It wasn’t a surprise to see Ramsey developing into his role - I trained for two years with him and saw him develop. He’s a great player and we have big expectations for him.
“We had a couple of injuries and it was hard to keep up with Manchester City but we have a stronger squad and deeper team now, with more players to use, and hopefully not as many injuries individually.”
Arsenal will have to do without right back Bacary Sagna, however, who has moved to Manchester City after seven years at the Emirates Stadium.
The club also passed up the chance to bring Cesc Fabregas back to the club for which he made his first-team debut, with Cazorla’s compatriot signing instead for Chelsea.
“Obviously he’s is a big loss,” Cazorla added of Sagna. “He was a very important player on and off the pitch, a quality player who did a lot for us. But I hope that we can replace him effectively.
“It’s strange [to see Fabregas join Chelsea]. He's of Arsenal, but in football you can play for other teams. It's an opportunity for him and I'm happy for him.”
One man who will remain at Arsenal for the foreseeable future is Wenger, who enters his 14th season with the club after signing a new three-year contract.
“He's a great person, he’s always been there,” Cazorla enthused. “It’s great for us to know he's signed a new contract, and it's great for us to work with him.
“He was very important for me from the very first day. He's been crucial, every game I play, to develop me as the player I am today.”
At 29, Cazorla now ranks among the senior members of the Arsenal squad and hopes he can play a part in helping guide the club’s younger players though the upcoming season.
“I like to be a leader and take responsibility to look after the young players,” he said. “It’s a role I am comfortable in and one I want to develop more next season.”
Arsenal's new home and away replica kits and a full collection of fanwear are available for fans to purchase now online at www.puma.com and in many stores worldwide. The new cup kit will launch in-store and online on August 14.