The two clubs faced each other on Tuesday night at Anfield in the quarterfinals of the Champions League, with the second leg to come in Turin next week.
Liverpool won the game 2-1.
It is the first meeting between the clubs since 29 May 1985, when 39 fans were killed in rioting at the European Cup final in Brussels, Belgium.
Michel Platini, scorer of the only goal for Juventus in that final, and former Liverpool stars Ian Rush and Phil Neal took part in an emotional ceremony before the game to mark the anniversary.
"It's a big night for both clubs," Platini said. "The bad memories of that night will never go away but let's hope that tonight will help everyone make friends."
"The bad memories of that night will never go away but let's hope that tonight will help everyone make friends"
former Juventus star
Uefa, football's European governing body, urged fans of both teams not to reopen old wounds.
"What happened at Heysel in 1985 was a tragedy for all those involved, the two clubs, the fans, their families and football as a whole," Uefa chief executive Lars-Christer Olsson said.
"Tonight we pay our respects to those supporters and families who suffered loss on that day."
"However, football has made significant improvements and advancements since that very dark day, and I am hopeful that as well as acting as a remembrance, this game will be looked on as the final part of the healing process."
Most of the blame for the disaster was directed at Liverpool fans who charged at Juve followers, causing a stampede. A wall collapsed and fans were crushed. All but one of the dead were Juventus fans.
Although the two teams have not played each other since that day, efforts have been made by both clubs to build a friendship to heal the wounds.
The Liverpool Echo newspaper carried the names of the 39 dead on its front page on Tuesday above the headline "We're Sorry".
"I don't hate the Liverpool fans for what happened and life has to go on. But I'm grateful there are ceremonies to mark the tragedy and the loss of life so that, hopefully, it doesn't happen again"
Terry Wilson, a Liverpool fan who was convicted of involuntary manslaughter in the Heysel deaths, travelled to Turin last weekend to apologise personally to the father of one of the victims, Roberto Lorentini.
"For me it was only nine months out of my life," said Wilson, who served only a short spell of a five-year sentence in a Belgian jail. "But they're still suffering, more than I'll ever know."
Juventus fan Antonio Masi said it would be difficult to concentrate on the game.
"I wasn't born at the time but I heard so much about Heysel from my family and friends and saw TV pictures," Masi said. "I don't hate the Liverpool fans for what happened and life has to go on. But I'm grateful there are ceremonies to mark the tragedy and the loss of life so that, hopefully, it doesn't happen again."
The most important thing is that the memory of those who died be respected in a dignified way.
Before Tuesday's match - dubbed the Game of Friendship - Liverpool fans carried a banner bearing the names of the 39 dead across the field to where the Juve followers were seated.
Juventus fans were invited to wear armbands carrying the colours of the two teams - red for Liverpool and black and white for Juve.