Crashing out in the last 16 would be unthinkable for two clubs that were among the favourites when the draw for the knock-out stages was made in December.
"Those that have ambition are not afraid," Allegri said, after his side were drawn to play Atletico, with the first leg on Wednesday at the Wanda Metropolitano.
"It's like a final, we cannot put a foot wrong," Atletico boss Diego Simeone said.
Juventus and Atletico Madrid have reached 12 European Cup finals between them, with last season's game in Kiev the first in five years in which neither had featured.
Ronaldo on target as Juventus cruise before Atletico showdown
Atletico went on to win the Europa League instead, proving themselves too strong for Europe's second-tier competition, while Juventus were dumped out by eventual winners Real Madrid, undone by a controversial Cristiano Ronaldo penalty in the 98th minute.
It was Ronaldo's 10th goal in seven games against Juventus, having scored twice in the first leg and twice in the final in 2017 too. Juve beat Real in the semi-finals two years earlier but even then, Ronaldo managed a goal in each leg.
There was understandable relief, as well as joy, when the Portugese switched sides last summer and now, there is belief. Ronaldo has not finished in single figures for goals in the Champions league since 2011.
As Lionel Messi said, "They already had a good team and now they have Cristiano Ronaldo."
But with anticipation comes pressure. Juventus are on course to win an eighth consecutive Scudetto but this is the one trophy they crave.
Miss out again and it will be 23 years since they last lifted Europe's premier prize, this time with its most prolific player ever among their ranks.
Ramsey leaves Arsenal after 11 years to join Juventus
"We have to be wary of Atletico," Allegri said.
"And we have to find the right moment to score. It's crucial that we get a goal there, otherwise qualification will be very difficult," he added.
Atletico's own record signing last summer, Thomas Lemar, was not even born when Gianluca Vialli hoisted the trophy in 1996 for the Italians but expectation weighs heavy on them too.
Atleti have the chance to play a final at home this year, at the Wanda Metropolitano, and not only snatch the cup away from rivals Real but do it in the city they both call their own.
"Of course we are inspired by playing a final in our own stadium, in our own city, in front of our own fans," Simeone said.
"But it is pointless only to imagine it. The reality will be what we deserve," he added.
Simeone extended his contract as Atleti coach until 2022 last week and while few can question the remarkable progress made during his tenure, there are some that believe a ceiling has been reached.
They are out of the Copa del Rey and seven points behind Barcelona in La Liga, where back-to-back defeats earlier this month failed to capitalise on a rare Catalan dip.
Summer signings are yet to fully fire and there is scepticism about the January arrival of Alvaro Morata, not just because of his links to Real. When results waver, questions about Simeone's style also invariably follow.
"It is a style I like, it makes us win," Antoine Griezmann told UEFA on Monday.
Winning a trophy on their own patch would be even better.
"It's an extra motivation for us and for the fans. The road to get there will be tough but we'll give it our all," World Cup winner Griezmann added.
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