Referee Anthony Taylor showed each a straight red card for a high tackle during Everton's 2-1 win at the Boleyn Ground.
West Ham striker Cole walked for catching Leighton Baines, and Taylor evened up the numbers late in the game after visiting midfielder Gibson challenged Mark Noble.
Both would have served three-match suspensions if appeals to the Football Association had proved unsuccessful.
But Gibson will now be available for Sunday's Sky Live game against Chelsea, provided a thigh injury suffered against Wigan on Boxing Day does not keep him out.
And Cole is likely to lead his side's attack at Reading on Saturday.
Cole tweeted: "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!"