Ozil and Sanchez are both into the final years of their deals.
Sanchez came close to leaving and joining Manchester City on deadline day.
Although Monaco accepted the Gunners’ £92million offer for Thomas Lemar, the player snubbed the move, scuppering Sanchez’s chances of sealing a switch to the Etihad as Arsene Wenger didn’t have time to find a replacement.
The Gunners have had a far from ideal start to the campaign, winning just one of their three Premier League games.
Alexandre Lacazette and Sead Kolasinac from Lyon and Schalke were Arsenal’s only summer deals.
But Gazidis believes the Gunners sent out a strong message by clinging onto their prized assets.
He told the Daily Star: “It was a clear statement of our ambitions and should help refute suggestions we are just about the money.
“That is simply nonsense.”
Although Arsenal sold Alex Oxlade-Chamberlain for £35m to Liverpool, Gazidis believes the club is in great shape.
He said: “At the end of the window, reviewing all activity, we have once again improved the quality and depth of our squad.”
Arsenal host Bournemouth in their first game after the international break on Saturday (3pm).