It emerged on Thursday night that the 22-year-old South Korea international striker had made four Barclays Premier League appearances for the club earlier this season with the Black Cats unaware that his international clearance had not been processed.
The Wearside club, fined in December for the offence, later revealed the problem was an administrative error which was due to a "technical fault" and denied any wrongdoing.
The news was greeted with interest by Sunderland's fellow relegation candidates on Friday, but Poyet, who was not at the club when the error occurred, insisted all parties had to abide by the Premier League's decision.
He 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."
That is precisely what Poyet's Brighton had to do after Hartlepool unwittingly fielded midfielder Gary Liddle, who was actually suspended, against them in a 2-0 League One win for the north-east club in April, 2010.
The Uruguayan campaigned for Pool to be docked the three points and the win awarded to the Seagulls, although he was unsuccessful in the latter respect.
Ji played against Fulham, Southampton, Crystal Palace and Manchester United in the league and MK Dons in the Capital One Cup before it emerged that his international clearance, which Sunderland had submitted electronically along with the other relevant documents on his return from a loan spell with German club Augsburg, had not arrived.
The error was only discovered by chance in October and the club immediately notified the authorities.
A club spokesperson said: "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."
MK Dons have confirmed they sought clarification on the matter when they became aware of it, but will take no further action.
A statement read: "The club acknowledges that the Football League board dealt with the situation as they saw fit and will not be pursuing the matter further."
Cardiff manager Ole Gunnar Solskjaer was also philosophical, although Hull counterpart Steve Bruce, who as Sunderland manager signed Ji from Chunnam Dragons in June 2011, was surprised by the news.
He said: "That's pretty bizarre stuff, that it's only surfaced now and he's played in four games. They should be in serious trouble, I would have thought."
However, Press Association Sport understands that the Premier League too considers the matter, which a source described as the "lowest level of administrative error", dealt with.
Poyet now faces the task of putting a turbulent week behind him and concentrating on the task of trying to preserve the club's top-flight status.
Asked if the Ji saga might prove a distraction, he said: "I don't think it will affect the team. When too many things happen around a club, sooner or later it does affect the team, I believe that.
"I know people think it shouldn't but it does. Now I don't think we have too many problems.
"But this has come as a surprise. People now maybe know why I couldn't play him. I knew, you didn't.
"I don't have Twitter or whatever, but I can imagine some people love Ji and when he was not playing every week and not on the bench, they were probably killing me.
"But they didn't know. I hope now they understand why I couldn't put him on the bench."