The France international produced two cool finishes, either side of a Carlos Tevez strike, as the reigning Serie A champions cruised to victory in Rome.
The result means Juve have now gone 20 league matches against Lazio without defeat, while it also restored their three-point lead at the top of the table after Roma had momentarily moved level with a 2-1 win at Atalanta earlier in the day.
Juve coach Allegri was full of praise for Pogba after the game and believes the 21-year-old could go on to become one of the greatest players in the world.
"Pogba had a great game, but he has the quality to do even better and become the strongest midfielder in the world," he told Sky Sport Italia.
"He moved with the right tempo in attack and defence. It was his best performance of the season so far.
"The team did well, defended when necessary and I think we played fairly well with some good passing moves.
"We struggled a little in the first half and misplaced a few too many passes, but sorted that out.
"The lads worked hard and [Federico] Mattiello nearly scored when coming off the bench. It was important to get a win in such a difficult arena against a tough Lazio side, especially after Roma had won earlier."
Allegri welcomed influential midfielder Andrea Pirlo back into the side after the 35-year-old had missed the 7-0 thrashing of Parma last time out and was pleased to have the experienced playmaker back among the ranks.
"Andrea Pirlo hadn't trained with the team for 10 days, but he had a great game and when we have an extra man between midfield and defence it's normal he has more space and protection to pass the ball," he said.
"The team works together and the more the forwards do, the easier it is for the defenders. You have to keep the ball as far away from your own box as possible and you do that by attacking."
