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The South Korean forward caught Gomes in the 79th minute of Tottenham's 1-1 draw with Everton and was distraught to see the Portuguese midfielder suffer a fractured dislocation of his right ankle.
Members of the crowd nearby turned away in shock at the grotesque angle of Gomes' ankle while Son appeared to be left in tears over the incident.
Referee Martin Atkinson appeared to reach for a yellow card before producing a red which would have resulted in a three-game ban for the Spurs star.
However, that decision has now been overturned by an independent panel who upheld Tottenham's appeal.
The FA statement said: "Son Heung-min will be available for Tottenham Hotspur FC's next three domestic fixtures after an independent regulatory commission upheld a claim of wrongful dismissal."
A Premier League statement after the game had initially said: "The red card for Son was for endangering the safety of a player which happened as a consequence of his initial challenge.”
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The successful appeal will delight Mauricio Pochettino, who was surprised to see his player sent off for the challenge.
"It was clear it was never the intention of Son to create the problem that happened afterwards. It is unbelievable to see a red card," Pochettino said.
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Semi-finals, Day 15
Round 1 - Donington Park