Defender Gibbs was shown a red card in a case of mistaken identity during the first half of Saturday's 6-0 defeat to Chelsea by referee Andre Marriner, instead of Oxlade-Chamberlain.
The FA confirmed on Monday evening the sanction against Gibbs had been removed and transferred onto Oxlade-Chamberlain.
But the Gunners appealed this offence by claiming that the England midfielder had not in fact technically prevented a clear goalscoring opportunity as the shot from Chelsea's Eden Hazard looked to be going wide.
A statement from the FA said: "An Independent Regulatory Commission on Monday heard two claims from Arsenal, one of mistaken identity and the other of wrongful dismissal, both in relation to Saturday's game against Chelsea at Stamford Bridge.
"Further to the red card wrongly awarded to Kieran Gibbs, The Football Association has released the following statement.
"The commission ruled that the dismissal of Kieran Gibbs was a case of mistaken identity and transferred this to his team-mate Alex Oxlade-Chamberlain.
"Subsequent to deciding this matter, the commission then considered Arsenal's claim for wrongful dismissal in relation to Oxlade-Chamberlain.
"This claim was upheld, meaning Oxlade-Chamberlain will not serve any suspension with the standard punishment withdrawn with immediate effect."