The Turin giants become the first team to complete the league and Cup double for four consecutive seasons.
Without mentioning them by name, he then took aim at Napoli who were welcomed by around 10,000 people at Naples airport after their win in Turin, which at the time cut Juve's lead to one point with four games remaining.
Roma, in third place, needs just a point to ensure top European football next season as it holds a four-point advantage on fifth-place Inter Milan in the race for the top four with two games to play.
Captain Gianluigi Buffon was sent-off for his furious reaction to a last-minute penalty which dumped Juventus out of the competition on aggregate despite a win on the night.
"As I always say, winning is something extraordinary", Allegri was quoted as saying by Juventus' website.
'It's a wonderful feeling, as it's never easy to win'.
"It's a club that succeeds in remaining united through difficulties". Keeping level heads always make the difference, and you could see it in the key games. You reach the final objective with one small step at a time. If you get overexcited, you risk falling apart.
Roma, which had little to play for having been guaranteed a place in next season's Champions League by Inter's loss to Sassuolo on Saturday, were far more positive in the opening period but barely went close to testing its former goalkeeper Wojciech Szczesny. Lazio have 72 points, while Inter have 69 points.
Allegri's side also won the Italian Cup in the Stadio Olimpico on Wednesday with a 4-0 win over AC Milan.
Lorenzo Insigne had a chance to open the scoring for Napoli in the 12th minute after going through on goal but Sampdoria goalkeeper Vid Belec made a good save.