Sir Alex Ferguson and Brendan Rodgers hope the Hillsborough tribute before Manchester United's 2-1 win over Liverpool at Anfield will end vile chants by both sets of fans.
However, incidents after Sunday's Premier League match suggested some supporters had still to get the message.
This was Liverpool's first home fixture since an independent report absolved their supporters of blame for the deaths of 96 of their fellow fans during a 1989 FA Cup semi-final at Sheffield Wednesday's Hillsborough ground.
There were fears United fans would disrupt pre-match tributes to the dead, as well as the game itself, with taunts about the disaster.
They were largely well-behaved save one brief chant of "Where's your Munich song?"
But after the final whistle, when most Liverpool fans had left the ground, a few home supporters ran across the main stand near to where United followers remained on police orders and started making airplane signals.
This was in relation to the 1958 Munich air disaster, which killed 21 people including eight United players, the famed "Busby Babes".
Some United fans then responded instantly with chants of "Always the Victims" and "Murderers" in a reference to the fact Liverpool were playing during both the 1985 Heysel disaster in Brussels, where 39 Italian and Belgian fans lost their lives during a European Cup final against Juventus, and Hillsborough.
"What was done at the end I cannot comment on as I didn't hear or see anything," said Liverpool manager Rodgers.
"But it is on the field where competition should be and everything else, songs from Liverpool or Manchester supporters, any of us that have any human decency don't like to hear that.
"The fight will go on if there is a continuation of that but certainly at this club the work that has gone on in the last couple of weeks is something I am very proud of and the tributes today were fantastic."
Veteran United boss Ferguson, in charge at Old Trafford at the time of the Hillsborough tragedy, added: "Liverpool did a fantastic job today, the fans were terrific and I don't think there can be any complaint on that part.
"It demonstrates these two clubs can do things with unity."
United won the game thanks to a late Robin van Persie penalty after Rafael had cancelled out Liverpool captain Steven Gerrard's early second half goal.
But Liverpool played more than half the match with 10 men after Jonjo Shelvey was sent off in the 39th minute after lunging in on United defender Jonny Evans.
Defeat left Liverpool in the bottom three and without a league win under Rodgers, brought into replace sacked Anfield hero Kenny Dalglish.
But a defiant Rodgers said there was still time for the club to turn their season around as Arsenal did last term when a shaky start culminated in Champions League qualification.
"I remember last year Arsenal didn't get off to a great start," Rodgers said. "We are seeing some positive moments. I have got great confidence, the group have been absolutely fantastic.
"We are all behind it. Very soon our luck will change."