The Gunners are seeking a new manager for the first time since 1996 following Friday's announcement that long-serving boss Wenger will move on at the end of the season.
A number of familiar names have been linked with the job, including ex-Chelsea boss Carlo Ancelotti - who would welcome an approach - but Gazidis is in no rush and has yet to speak to any candidates regarding the high-profile position.
"The most important thing is to make the right appointment - not a quick appointment," he told reporters at a press conference. "The process begins today. I want to keep that process in-house. This club has to rise to the challenge. You don't find a replacement for Arsene Wenger, you find a new path forward.
"We need to be open-minded and also brave in the decision. When Arsene was appointed, I don't think he was on many people's radar. That doesn't mean we have to repeat that but it does mean we have to be bold in the appointment. Get the person we believe is the right person.
"It's important to me that we continue the football values that Arsene Wenger has instilled - somebody who can continue that for our fans - progressive, exciting football that gets people interested and excited in games we play.
"Another significant piece is how the candidate represents the club and our young players - our academy teams are doing extremely well and under the radar there are lots of players coming into our first team. Tomorrow we have to start thinking about how we move forward. We are going to take what Arsene has given us and take that forward. We have to go through that process."
Wenger's next game comes at home to West Ham United in the Premier League on Sunday afternoon.