Fellow Bombers fans reported the abuse, and the offender was reported to police.
Essendon says it has contacted Goodes to apologise.
Chief operating officer Xavier Campbell said the Dons would not tolerate racial vilification in any form.
"Racial vilification is completely unacceptable and will not be tolerated," Campbell said.
"As a club, we do not accept appalling behavior of this nature - No matter your race, religion or gender, football is a game of inclusion.
"It is extremely disappointing incidents of this nature are still occurring.
"The member in question has expressed his deep remorse for the incident and accepted full responsibility for his actions."
"His membership has been terminated and his reapplication for membership will only be considered after he has undertaken a racial vilification education program."
Goodes, who was not aware of incident until after the game, said he wished to thank the Essendon Football Club for their handling of the issue.
“While it is disappointing that these incidents still arise, there is a positive to come out of this, and that is the willingness of people in the crowd, regardless of the colours of their scarf and jumper, to draw a line in the sand and say that this is not acceptable," Goodes said.
“While there still work to be done, to have Essendon members alert stadium security to the incident is a great indication that people in the football community will not tolerate racial vilification.
“I look forward to the up-coming Indigenous Round, where we can once again celebrate the contribution of Indigenous players in our game," he said.
The news comes after Bombers were slammed on social media for cheering when umpire Troy Pannell suffered a serious head collision in Friday's match, and then booing as he was stretchered from the field.
Meanwhile, the Western Bulldogs have apologised to Melbourne player Neville Jetta after a man wearing a Dogs jumper was heard shouting racist comments during Round 8.
The Bulldogs are still trying to find the offender.
"It has been the club’s initiative to make this matter public and to acknowledge the wrongdoing of one of our own supporters," president Peter Gordon said.
"When racist abuse rears its ugly head we will never as a club do what this person has done...disappear into anonymity. It happened. One of our own supporters did it.
"We are all diminished as a consequence. But we will not let it pass," he said in a statement.
“It is appalling that some people still think they can go to the football and yell whatever they like over the fence,” Mr Evans said.
AFL football operations boss Mark Evans said the strong response from supporters in the surrounding area who reported the comments and the subsequent actions of the clubs involved demonstrated that vilification and anti-social behaviour have no place in Australian Football.
"The AFL has been very clear that vilification of any kind is not acceptable at football matches and, whenever possible, action will be taken against those responsible,” Evans said.
"While these two isolated incidents are very disappointing, it is good to see that AFL crowds are now largely self-regulating when it comes to offensive comments and anti-social behaviour."