Juve reached the last eight courtesy of a superb 5-1 aggregate victory over Borussia Dortmund, including a comprehensive 3-0 triumph at Signal Iduna Park on Wednesday.
The Turin club have faced criticism in recent times for failing to replicate their domestic domination - which has seen them win three straight Serie A titles - in the Champions League, having been eliminated at the group stages in 2013-14.
However, Juve director Nedved is confident the team are a forced to be reckoned with under Allegri.
"We need to congratulate the boss who has equipped the team with the mentality to tackle European games," he said. "He and his staff are doing an incredible job and he's managed to transfer his expertise over to the players.
"Now we need to stay on the ball for the rest of the season. We're without a couple of important players who are sidelined through injury and the league's not done and dusted.
"We need to face every game with the utmost focus, which is vital if you want to go all the way. Any other mentality would be wrong."
Juve start as favourites against Ligue 1 side Monaco, who defeated Arsenal on away goals in the last 16, but Nedved warned against complacency.
"You don't just get to this stage of the Champions League by chance, Monaco put in two great performances against Arsenal and particularly impressed me at the Emirates Stadium," the Czech added.
Meanwhile, Monaco head coach Leonardo Jardim concedes that his side will be considered underdogs for the two-legged tie, but feels that playing the second leg at home may prove crucial.
"Juventus are among the biggest sides," he said. "They are the favourites.
"We're lucky to have the second leg at home. We'll try to make the most of this quarter-final."
Get the best features, fun and footballing frolics straight to your inbox every week.
Thank you for signing up to Four Four Two. You will receive a verification email shortly.
There was a problem. Please refresh the page and try again.