After departing Milan midway through the previous campaign following a run of poor results, Allegri was appointed as Juve boss in July after Antonio Conte's surprise exit.
Allegri began his reign - and Juve's title defence - with a 1-0 win at Chievo on the opening weekend of Serie A, and he watched on at Juventus Stadium on Saturday as goals from Carlos Tevez and Claudio Marchisio sealed a routine victory.
The coach was delighted with the result, although he admitted there were elements of their game that his players will have to work on.
"It was a special night for me as Juventus coach in this wonderful stadium," he told Sky Sport Italia.
"I was curious to see how the night would go, above all that we'd be able to win!
"The crowd is an added weapon for the team and they stepped up when there were moments of difficulty.
"We had this 10-minute spell when Udinese had us under pressure and we've got to emerge from those situations better. We also gave the ball away in avoidable ways.
"The team started very well and we knew Udinese would become more dangerous on the counter as time wore on, so had to break the deadlock early.
"The lads defended well and once we took the lead we were at times patient, at others a bit too hasty. I wanted more than one goal and we suffered in the second half, unable to get the measure of Udinese after their tactical switch, but in the end it was a good performance.
"I am absolutely not surprised we've started the season so well. This squad has physical and technical qualities which mean we can do a lot better.
"We must improve on how to control the tempo, as we cannot always go at 100mph or at 10mph. Playing every three days we have to learn more control to avoid wasting energy."
