Juve beat Napoli 3-1 on Sunday to gain revenge for last month's Supercoppa Italiana defeat to Rafa Benitez's men and pick up their first league win of 2015.
Late goals from Martin Caceres and Arturo Vidal ensured maximum points for Juve, who had seen Paul Pogba's opener cancelled out by Miguel Britos at the Stadio San Paolo.
While Allegri was delighted to see his side come through a close-fought affair with a victory, the former Milan boss called for more from his charges.
"Leagues are won by coming through difficult games, showing the ability to defend in a certain way, giving little away and playing well from a technical standpoint," he explained.
"The fact that Juve have won the last three Scudetto titles does not mean that this year will be straightforward and we must improve.
"We're often good at getting plenty of bodies into the opposition half and winning the ball back quickly, but we need to work more on our distribution of the ball, balance and preventing shots coming in on our goal."
Defender Leonardo Bonucci was more generous in his praise of Juve's win, believing that it sent a message following their tricky recent form.
"We required a reaction after the Supercoppa defeat and recent draws against Inter and Sampdoria," he added.
"[Against Napoli] our response was emphatic, striking at the right moments. We showed that we still have our spirit and proved any doubters wrong.
"Winning and showing the right spirit was what mattered to us. Often being aware of your strength leads you to being overconfident, but we showed that we're a great team."
