FA rescinds Gibson, Cole red cards
Both players will now see potential three-match suspensions written off.
Everton lodged their appeal with the Football Association on Christmas Eve, meaning Gibson was able to face Wigan on Boxing Day.
Cole will be available for West Ham's Premier League match against Reading on Saturday.
Cole wrote on Twitter: "Great news that the red card got overturned, @FA done a great job. Well done! Time to make all the aggro worth it on Saturday if selected!"