"Madrid have a lot of strengths. They're an extraordinary side with great technique and pace. They're used to playing in these kinds of games, so it will be a great final," Allegri told UEFA.com on Thursday.
"We should be very pleased that we've made it this far, but we're going into this year with a completely different level of belief than in 2015.
"Compared with the 2015 final in Berlin, our squad has changed in almost every position.
"Only four or five players are still here. Our confidence has increased, our awareness of our own ability has gone up, the atmosphere has improved.
"This time we have to lay our hands on the trophy."
Allegri's side became the first team to win Serie A six times in a row last month and should they beat Real they would land the treble after also winning the Copa Italia.
"We have raised the bar after winning the Scudetto six times in a row, reaching the Champions League final, winning the Coppa Italia twice and the Supercoppa," Allegri said.
"This team needs to constantly set the bar even higher and go after an even bigger target."
While Juve's defence, built on the rock-like central axis of Giorgio Chiellini and Leonardo Bonucci, is at the heart of everything Juve have achieved this season, Allegri's side have added potency up front since a tactical tweak this year.
Having been a disciple of the 3-5-2 system, Allegri has shown a liking for a 4-2-3-1 system that allows him to fully utilise the attacking weapons of Gonzalo Higuain, Paulo Dybala, and Mario Mandzukic while at the same time allowing Juan Cuadrado to thrive out wide.
With Brazil's Alex Sandro a key link in midfield with his tackling ability and eye-catching passing, it's a system that allows Allegri to fight fire with fire in Cardiff on Saturday.
"The 4-2-3-1 formation came to me the moment I realised our team wouldn't progress any further if we kept the old tactics and formation," Allegri said. "I assessed our players' strengths and I tried to put them on the pitch in their favourite positions -- and I also tried to encourage our attacking skills.
"It had a lot of beneficial effects: our players enjoyed themselves a lot more on the pitch.
"Having a lot of attacking players on the field and a good balance at the back is the foundation of victory."
(Reporting by Martyn Herman, editing by Pritha Sarkar)
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