Lewis, who was appointed Executive Vice-Chair in March 2023, has been advising Kroenke Sports & Entertainment (KSE) since 2007 and joined Arsenal's board as Director in September 2020.
Garlick, who arrived at Arsenal in 2021 as Director of Football Operations, has served as Managing Director for the past year.
"Rich has made an enormous impact on all fronts as we continue to strive to win major trophies, be financially sustainable and put our supporters at the heart of everything," Arsenal Co-Chair Josh Kroenke said.
"At this moment, we would like to thank Tim for his continued dedication and commitment to Arsenal in a period of transformational change for the club.
"He has played a pivotal role and ensured we are in a great position to continue to deliver our strategy in our ambition to win major trophies."
Arsenal have finished runners-up in the last three Premier League seasons, with their league title drought now 21 years.
They again recruited heavily in the close season, adding the centre-forward fans have long called for in Sweden's Viktor Gyokeres, bolstering the attack with England's Noni Madueke and Eberechi Eze and signing Spanish defender Cristhian Mosquera.