Pogba joined United from Juve in a world-record €110 million deal in the summer after four Scudetto-winning seasons in Turin.
The Bianconeri have reclaimed top spot this term but lead Roma by only a point - albeit with a game in hand - and have lost four games out of 19 after being defeated just five times in the entirety of the 2015-16 Serie A campaign.
Their latest defeat came at the hands of Fiorentina on Sunday, with goals from Nikola Kalinic and Milan Badelj downing the defending champions despite Gonzalo Higuain's reply.
Chiellini believes the difference is largely down to Pogba's absence, claiming the Frenchman exerted an influence from the centre of the park that allowed Juve to dominate games more easily.
"Last year we had a player like Pogba in midfield, who is the LeBron James, the Usain Bolt of football," Chiellini told Sky Italia.
"Last season Paul allowed us to play a certain way, but now he's not here and we made some changes. [Massimiliano] Allegri is working on finding a solution, we aren't idiots or clowns.
"We're giving up too many chances - last season [Gianluigi] Buffon never dirtied his gloves."
Pogba has made 20 appearances in the Premier League for United this season, scoring four goals in the process.
Juventus brought in Miralem Pjanic from Roma in the summer to replace the 23-year-old and have added Tomas Rincon, from Genoa, to their midfield ranks this month.