The Black Cats are locked in a desperate battle for Barclays Premier League survival along with the likes of Fulham, Cardiff, West Brom, Crystal Palace and Norwich.
However, their plight - they went into the weekend four points adrift of safety - could have been significantly worse had they seen points taken away after inadvertently fielding striker Ji without international clearance earlier in the season.
But head coach Poyet, who campaigned unsuccessfully for his former club Brighton to be awarded the three League One points docked from Hartlepool following a 2-0 defeat for the Seagulls, in which midfielder Gary Liddle played when he should have been serving a suspension, is adamant all parties have to abide by the decision to fine them instead.
The Uruguayan said: "It's down to the decision. I haven't forgotten the Tevez-Mascherano situation with Sheffield United.
"We can talk, but at the end of the day, the one who makes the decisions made one and here we are."
Hull boss Steve Bruce, who signed Ji from Chunnam Dragons in June 2011 during his time as Sunderland manager, voiced the opinion that his former employers could be in "serious trouble", although Cardiff counterpart Ole Gunnar Solskjaer was more philosophical.
Solskjaer said: "We accept whatever the outcome is. I don't say I'm not bothered, but there's not a lot I can do."
However, the Black Cats remain adamant the problem was the result of a technical error rather than any wrongdoing on their part.
Ji was registered as their player and they submitted his international clearance electronically along with other documents following his return from a loan spell at German club Augsburg.
However, it never reached the Premier League and the omission was not noticed until October, when the club immediately notified the authorities and subsequently accepted the punishment.
A club spokesperson said on Friday: "The issue of Ji Dong-won was explained in detail at today's Premier League shareholders meeting and no issues were raised by any club.
"Sunderland AFC confirmed that they had complied with all procedures and the Premier League stated that the player was duly registered on the extranet system and in all other Premier League mechanisms.
"The club has never accepted any wrongdoing, but did acknowledge that a technical fault occurred and as such were fined by the Premier League accordingly in December. The matter is now closed."
Poyet, who took over as manager days after the issue arose, was made aware of the situation, but all parties opted to keep it private.
The head coach said of Ji, who completed a permanent move to Augsburg in January: "For me, he couldn't play, so I can only talk about him as a professional. He was outstanding.
"He understood the situation. He waited, he didn't say a word, so as a professional and as a person, Ji left this football club with everybody being on his side because he deserved plenty of credit for what happened to him and how he coped personally with the situation."