Vermaelen recently completed a move to Barcelona, with Sagna joining Premier League champions Manchester City earlier this summer.
But Gibbs is not fazed by their exits, suggesting that new men Calum Chambers and Mathieu Debuchy can step up and replace them for the Gunners.
"Calum Chambers and Mathieu Debuchy have also started impressively," he wrote on Sportlobster.
"Obviously, the quality of Bacary Sagna and Thomas Vermaelen was always going to be missed but Calum and Mathieu look to have the ability to step into their boots."
Meanwhile, the 24-year-old has expressed his excitement ahead of the new Premier League season.
Arsenal host Crystal Palace on Saturday afternoon and believes the Gunners' Community Shield victory over Manchester City was a sign of intent.
"I am incredibly excited for the new season, because we have made important signings and the boys are feeling positive after wins in the FA Cup Final and Community Shield respectively.
"We hope our win against Manchester City in the Community Shield is a sign of intent."