The Frenchman will leave the club at the end of the season, bringing to an end his near 22-year reign as manager.
Wenger, 68, won three Premier League titles and seven FA Cups, including the Double in 1998 and 2002.
“The most important thing is to make the right appointment – not a quick appointment,” said Gazidis.
“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.”
Gazidis on the future
Regarding Wenger’s replacement, Gazidis said Arsenal do not want to lose the “qualities and values” the Frenchman has brought to the club over the past 22 years.
“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.”
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On Wenger’s legacy
Gazidis said Wenger made Arsenal famous with the style of football he instilled in his teams, and inspired people he worked with through his “understanding and empathy”.
“Arsene earned his position at this club over 22 years. I don’t think there will be other managers who will have that authority at the top of the game at the biggest clubs.
“It’s the most important appointment any club makes and still the most important employee any club has. It would be unrealistic to think anyone comes in with the breadth and scope Arsene has at Arsenal Football Club.
“Arsene changed the game, set a totally new standard, a new ambition, not just to win but to win while achieving perfection – to make art out of football. He was always brave enough to be true to that ambition and achieved it with the Invincibles.
“Arsenal truly has a football philosophy and he has been brave enough to live by it – reinvented the game and played some of the best football we’ve ever seen.
“The decision had to happen at some point. I don’t think the club has ever been better prepared for that decision. Now it’s time to look forward, and I think we can do that with confidence.
“He leaves us in a very strong position to look forward.”
On Wenger announcing his departure now
The chief executive said the timing of Wenger’s decision gives both parties time to plan for the future.
“Our first priority is to come together as a club. Look at the outpouring of affection from the Arsenal family and the football family.
“Secondly, Arsene and the club need time to prepare. It’s going to be a condensed summer with the World Cup in the middle of it. Having clarity sooner is better.
“I have often said replacing Arsene is the biggest challenge we face. In the coming months the world will see the unity and power of this football club, and the people within it.
“People are talking wonderfully about what Arsene has done for Arsenal and the wider world of football. His achievements are extraordinary and many will not be repeated.
“There is a widespread affection from Arsenal fans and people right across sphere of football. He will get a send-off the world will take notice of.”