The incident happened during the third quarter of the Power's 69-point victory over the Bombers at Adelaide Oval during Gather Round.
"I take full responsibility for what happened Friday night," Finlayson said in a statement released by Port Adelaide on Sunday.
"The word I used is very unacceptable in the game of football. We need to stamp it out and I'm very remorseful."
It is understood an umpire's microphone picked up the insult and multiple players heard it.
Port Adelaide said Finlayson, who kicked three goals from 19 disposals against the Bombers, made the club aware of the remark during the three-quarter time break.
He spoke with the victim on the field to apologise after the final siren.
"I knew straight away that it was not acceptable and I take full responsibility," Finlayson said.
"I addressed it at the time and ... let everyone know what happened, and it's now in the hands of the AFL to investigate.
"I'm continuing to reflect and improve myself, getting all the education I can to make myself better."
South Australia premier Peter Malinauskas said it was a credit to Finlayson that he had owned up to his error.
"People make mistakes, we all do," Malinauskas said on Sunday.
"It's a great credit to Jeremy that he's apologised quickly, he owned up to the mistake he has made.
"That's the action of someone who has done the wrong thing and acted with integrity in following.
"Obviously people will make commentary on how disappointing it is, but here we have a young man who has made a mistake and owned up to it."
The AFL Integrity Unit's investigation is ongoing.
"We've got to run a process, speak to all involved, but Jeremy's come out, he's admitted making the comment and he's apologised," AFL chief executive Andrew Dillon told Channel Nine.
"So now we've just got to work through what the accountability will be.
"I don't want to put too much of a time frame on it, but it will be done this week."
AFL CEO Andrew Dillon expects to move on the Jeremy Finlayson homophobic slur this week. (Joel Carrett/AAP PHOTOS)
Last month, North Melbourne coach Alastair Clarkson was fined $20,000 and has a suspended two-match ban for abusing St Kilda defenders Jimmy Webster and Dougal Howard during a pre-season game.
Clarkson was asked to explain his actions to the AFL after allegedly using the homophobic slur "c*******er" during the practice match.
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