Oxlade-Chamberlain used his hand to divert a shot in Saturday's Premier League match but referee Andre Marriner sent off left back Gibbs in a case of mistaken identity despite Oxlade-Chamberlain owning up to the foul.
The FA said in a statement that not only would Gibbs' offence be transferred to Oxlade-Chamberlain but that the case was one of wrongful dismissal.
Arsenal F.C. had argued the ball would have gone wide if the midfielder had not handled and therefore did not constitute denying a goal.
"This claim was upheld meaning Oxlade-Chamberlain will not serve any suspension with the standard punishment withdrawn with immediate effect," the FA said.
Despite his double error, the Premier League said on Monday that Marriner would not be demoted and would take charge of Southampton against Newcastle United on Saturday.