Chiellini scored all three goals at the Stadio Renzo Barbera, with the Juve man ultimately settling the Group H contest in his side's favour eight minutes from time.
The experienced defender had opened the scoring just prior to half-time, before putting through his own net with 13 minutes left.
With Antonio Conte's 100 per cent record as Italy coach looking perilous, Chiellini headed home the winner to ensure he made headlines for the right reasons.
"It would've been a real shame [not to win], as we risked compromising a victory that was fully deserved," he told Rai Sport.
"Considering how we've started this new era, it was only right to continue winning and the performance deserved it.
"We are hard-working and enthusiastic. We made some mistakes in our finishing today, but had the right attitude."
Asked about his own goal, which came about under pressure from Rauf Aliyev inside the area, the Juve man added: "The ball emerged from nowhere and I couldn't get out of the way.
"Oh well, I did everything tonight, good and bad."
