The pair shared details of a series of messages they received via social media on Sunday night.
Their clubs have strongly condemned the messages, with the Saints and Power confirming on Monday they were working with the AFL Integrity Unit to identify those responsible.
Wanganeen-Milera received messages from an anonymous account after the Saints' narrow five-point loss to Sydney.
The messages appeared to be related to betting on Wanganeen-Milera's disposal tally.
The 22-year-old playmaker had 24 disposals under close attention from Swans tagger James Jordon.
In response to the abuse, Wanganeen-Milera posted: "Over a game of football! Hero mate."
He received support from Saints teammate Liam Stocker, who reposted Wanganeen-Milera's story with the message: "Get a grip. Pissant. Put your name on it you donkey."
"As a club we do not accept racist behaviour of any kind," Saints CEO Carl Dilena said in a statement.
"These comments are abhorrent and have no place in our game or our community.
"Beyond Nasiah's talent on the field, he is a proud Kokatha and Narangga man who deserves to work and live free from this type of abuse.
"St Kilda Football Club stands firmly with Nasiah, his family and all First Nations people, and we remain committed to fostering a safe and inclusive environment for everyone."
Burgoyne also shared an image of messages he received, which also targeted his family, after Port's 26-point win over West Coast.
The 21-year-old posted the screenshot with a clown emoji seemingly directed at the troll.
"Such abuse is reprehensible and will not, and should not, be tolerated," Port Adelaide said in a statement.
"The club has advised the AFL integrity department and we look forward to working closely with the AFL to identify those responsible and sanction appropriately.
"We also encourage people who witness or experience online abuse to report and highlight it to the social media platforms on which it appears."
Port Adelaide also called on governments to demand accountability from social media organisations.
"Platforms need to take accountability and action against these abusers," the club said.
The AFL and AFL Players Association have repeatedly condemned online racial abuse of players in recent years.
AFL CEO Andrew Dillon acknowledged the courage of Wanganeen-Milera and Burgoyne in calling out the "unacceptable" racist comments, and offered the league's support to the pair.
"To hide behind a fake account is cowardly and I want the people that have created these online accounts and racially abused our players to know that our game does not want you," Dillon said.
"Real fans don't racially abuse players.
"We have been clear that there is absolutely no place for this behaviour in our sport and in society.
"Racism is never acceptable and these incidents demonstrate there is so much more work to be done."
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