Roberto Mancini felt his Italy side should have seen Poland off well before the stoppage-time winner which earned a narrow 1-0 victory in Chorzow.
Cristiano Biraghi struck in the second minute of added time to earn the Azzurri victory and relegate Poland from UEFA Nations League League A Group 3.
Lorenzo Insigne's corner was flicked on by Kevin Lasagna and Biraghi turned the ball in at the back post. It was no less than the visitors deserved after 18 attempts at goal, with Jorginho and Federico Chiesa both denied by the crossbar in the first half.
Mancini said on the UEFA website: "We dominated the game completely and should have scored earlier. A 0-0 draw would have been an unfair result."
The win means the Azzurri can still reach the finals ahead of Portugal, their opponents next month.
Mancini added: "We played very well but we can still improve a lot. In football you only need time and hard work, magicians do not exist.
"We need more time to build a team, but I am very pleased with today's game and result. There's a long way ahead of us, but the most important thing is that the players show tonight's mentality in the upcoming games."
Biraghi, the Fiorentina captain, dedicated his goal to Davide Astori, who died in March after a heart attack. He was 31.
Biraghi said: "Astori is a part of me. I dedicate my goal to him because if I'm here it is only thanks to him and all the things he taught me when we played together.
"The victory was fully deserved even if the ball didn't go in after all the chances we created.
"After missing so many opportunities, you can easily lose the game because the tension can be tricky. We never give up, so this victory belongs to the whole group."
Poland had goalkeeper Wojciech Szczesny to thank for keeping Italy at bay, but coach Jerzy Brzeczek was left to rue the defeat.
Brzeczek said: "I'm disappointed that we've been relegated from the top division. This is a difficult moment for us.
"These two games showed how much work is ahead of us. We tried to play in a different system against Portugal and Italy, but we saw that it's not for us.
"This game against Italy was the worst of the four games I've been in charge of. Szczesny saved us from conceding more goals."