Jorginho netted from the penalty spot to secure a point for the Azzurri after Piotr Zielinski opened the scoring for the visitors at Stadio Renato Dell'Ara in Bologna.
The game was Italy's first competitive match under new manager Roberto Mancini, who was appointed in May following Gian Piero Ventura's failure to reach the World Cup.
With Italy only taking a point against Poland, captain Chiellini warned that it would take more time to rebuild and must continue to improve their performances.
“We made a few too many errors and at the end of the day the only chances Poland had all came from our mistakes,” Chiellini told Rai Sport.
“Poland are a solid side who wait for you to get it wrong so they can counter-attack. We had a decent game and could’ve taken the lead with Federico Bernardeschi, but then they scored on a fantastic Robert Lewandowski assist.
“We must build on the second half performance, as we are still trying to construct a new identity for this Italy side. We’ll look back over the errors, without pinning the blame on anyone, and try to work on improving ourselves.
“We are all here to work, grow and achieve positive results in that process.”
One of Mancini's changes was at the back where he handed Cristiano Biraghi his international debut deputising at left-back for Domenico Criscito.
The Fiorentina defender admitted that Italy needed some time to build composure but was happy with how the team gained focus as the match progressed.
“We were a little too hasty during the game and that led to some misplaced passes," Biraghi told Rai Sport. "Once we were sharper and more focused, we created some good scoring opportunities.
“At the start, Poland’s attitude of sitting back and waiting to counter-attack unnerved us. That was their objective and they man-marked Jorginho constantly, so the central channels were blocked off. It meant we had to do more going forward on the wings.”