United's biggest win of the season was marred by a short burst of anti-Liverpool chanting from a small minority of home fans following this week's damning report on the handling of the Hillsborough disaster in 1989, in which 96 people died.
The report laid bare a shocking cover-up which attempted to shift the blame for the tragedy on to its victims, revealing that South Yorkshire Police had instructed officers to change or amend their statements relating to the tragic events.
One clearly audible chant at Saturday's game at Old Trafford was "Always the victims, never your fault", believed to be in reference to the report findings.
However, the song has been sung since Liverpool striker Luis Suarez was found guilty of racially abusing United defender Patrice Evra last season, which saw the Uruguay international given an eight-game ban and a £40,000 fine.
It is understood the Premier League deem the chants to be unacceptable but believe it is a matter for the clubs themselves to deal with.
There is a history of sour relations between the two sides. A minority of United fans have directed chants to Liverpool fans regarding Hillsborough and a Liverpool minority have chanted in reference to the Munich air disaster of 1958 in which 23 people died.
A United spokesperson said the club "deplore" such songs and, in a statement, the club said: "The manager has made the club's position very clear on this matter. It is now up to the fans to respect that."
The Football Association declined to comment on the matter on Sunday.
The tension between the two rival clubs is sure to be heightened when United travel to Anfield next Sunday.
Manchester United's Supporters' Trust echoed Ferguson's statements.
MUST chief executive Duncan Drasdo said: "We enjoy a fierce rivalry but these issues transcend that rivalry.
"We agree 100% with the statement made by Sir Alex Ferguson - this is the time for supporters of these two great clubs to represent their clubs with the integrity and honour that our glorious history demands.
"Following this week's developments and release of revelatory information on the Hillsborough tragedy, MUST wishes to make it absolutely clear that just as we condemn chants mocking the Munich air disaster we also condemn any chants relating to Hillsborough or indeed any other human tragedy.
"We did hear the usual anti-Liverpool chants at the match today [Saturday] but we're pleased to say, despite some reports to the contrary, there was nothing that was specifically referencing Hillsborough."