The midfielder, 29, was dismissed in the 24th minute of the Carabao Cup semi-final first leg for a reckless waist-high tackle on Diogo Jota.
The Gunners held on to secure a valuable 0-0 away draw but Xhaka felt he let down his side.
And on Instagram after the match he expressed his remorse, writing: "I want to apologise to everyone.
"I am so proud of my team and the spirit that they showed to keep a clean sheet tonight."
Rob Holding replied immediately, saying: "Together". The defender also posted two emojis - a heart and a muscle to show he was backing his team-mate.
Stand-in captain Alexandre Lacazette wrote similar: "We are together."
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