Juve suffered defeat in their last Champions League encounter, succumbing to a 1-0 loss at Atletico Madrid in a result that marked their fourth reverse in five away games in Europe's premier club competition.
The Turin club have never lost to Olympiacos on Greek soil but that record could well be under threat against a team that beat Atletico 3-2 in their opening Group A encounter and have put the likes of Manchester United and Arsenal to the sword at the Karaiskakis Stadium in recent times.
Former Milan coach Allegri is expecting an intimidating atmosphere in Piraeus and believes Juve will need to stay calm in order to have a chance of taking maximum points from the clash.
"Olympiacos have quality players and we'll have to play with intensity to try and limit their best attributes," Allegri said.
"We'll need to keep our heads in a heated atmosphere tomorrow and put in a good performance."
Group A was expected to be dominated by Atletico and Juve, however, all four teams are level on three points after the first two fixtures.
Juve suffered elimination in the group stages of last season's tournament but, although Allegri accepts that a victory in Greece would be important, he does not view the result as pivotal to their hopes of avoiding the same fate this time around.
"Tomorrow is important, not decisive," Allegri said.
"We need to come out of the two games [against Olympiacos] with as many points as possible in order to go through."
'People will remember us because we got to No.1 in the world rankings and achieved great things - but our biggest regret is that we didn’t win anything': Ex-Everton star Kevin Mirallas opens up on 'honour' of being part of nation's golden generation
‘I’m sure we’d have finished in the top four or five if he hadn’t joined Chelsea in January. Him leaving damaged the team, but we couldn’t stand in his way’: Alan Curbishley reflects on failed Champions League dream after selling key player
