The Hoops are set to face their first season in the Clydesdale Bank Premier League without Old Firm rivals Rangers.
Ally McCoist's men will play in the Irn-Bru Third Division - providing they gain Scottish Football Association membership - after the Ibrox club were consigned to liquidation.
Despite the dramatic changes facing Scottish football, Lawwell is confident Celtic can cope with what lies ahead, especially with the backing of supporters.
In an open letter to fans, he said: "As we embark on our 125th season we enter a crucial new phase in the great history of our club.
"Your unwavering support throughout the years has been central to the success of Celtic. It has been one of the core strands of Celtic's DNA.
"There have been great times throughout our history, none more so than Lisbon in 1967 or our centenary year in 1988.
"There have been challenging times too and following recent events, Scottish football and Celtic FC now finds itself on the verge of a new era.
"Throughout our history Celtic has, time and again, stood up for what it believes is right. In recent weeks we have acted accordingly, in the best interests of Celtic and Scottish football.
"It is clear that recent events will present the club with some new challenges - we are not afraid to face these head-on, with our supporters side by side.
"We have a club which is strong, a club open to all, formed with charitable purpose at its core, a club whose traditions and values unite millions of people across the world and a club which has enjoyed 125 unbroken years.
"More than ever, it is a time to celebrate what we stand for.
"We must do so with a collective unity between the supporters, the club and the team. We need to stand shoulder to shoulder and work together to continue to move the club in the right direction."
The letter continued: "Celtic supporters have already renewed their season tickets in huge numbers.
"I want to thank you all sincerely for your support. You are the lifeblood of the club and it is wonderful that you stand with us at this very important time in our history.
"For those fans still considering renewal or whether to buy a season ticket for the first time this year - we need you back at Celtic Park.
"It is imperative that we continue to develop this exciting young team and give them a platform to take on the best in Europe.
"Our priority is to give Celtic every opportunity to be as successful as possible, both domestically and in European competition.
"It's an ambition that every Celtic supporter, including myself, shares. We can only do that with your support.
"Everyone at Celtic sincerely understands and appreciates the commitment and sacrifice which you make in following our great club.
"One hundred and twenty-five years ago Brother Walfrid established Celtic Football Club.
"Celtic greats such as Willie Maley, Jimmy McGrory, Jock Stein, Billy McNeill, Tommy Burns and Henrik Larsson shaped our history.
"We now need to make our own history. Let's do it together."