The Turin giants had needed just a point to officially seal their 34th Scudetto title and become the first team to complete the league and Cup double for four consecutive seasons.
Allegri's side also won the Italian Cup in the Stadio Olimpico on Wednesday with a 4-0 win over AC Milan.
With one game left to play Juventus cannot be caught by second-placed Napoli who are four points behind after winning 2-0 at Sampdoria on Sunday.
"It's not easy to keep the right mentality just a few days after winning the Italian Cup final," said Allegri.
"I think the greatest strength was keeping calm in all situations. You reach the final objective with one small step at a time.
"It's been really tough and we have to compliment Napoli, as they kept the pressure on to the end and won at Sampdoria.
"We work hard, but when it's time to celebrate after four extraordinary years, it's right that everyone celebrates.
"If you get overwhelmed by enthusiasm, you're going to collapse three days later." Eusebio Di Francesco's Roma, in third, had already sealed their Champions League place next season after Inter Milan's 2-1 loss against Sassuolo.
"Juventus have a special mentality that has been built up through the years," said Di Francesco.
"It's a club that succeeds in remaining united through difficulties. They are an example for other teams to follow."
Juventus had early chances with Edin Dzeko firing over, a Lorenzo Pellegrini right footer flying wide and Aleksandar Kolarov's free-kick hitting the side netting.
Dybala threatened early and broke through early in the second half after an Alex Sandro cross but the goal was disallowed because the Argentina striker was offside.
Nainggolan received two yellow cards in the space of five minutes and from the 68th minute Roma were down to 10 men.
But Juventus did not take advantage of their advantage as they game lost its momentum although Dybala did have penalty appeals.
Juventus will play their final game in Turin next weekend and Allegri promised a big celebration.
"There are players here who won all seven titles and deserve Saturday's salute in front of their home fans. We want to celebrate all together in Turin," added Allegri.
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