Napoli forward Insigne had also scored against AC Milan on Saturday, fresh from being left on the bench as Italy failed to qualify for the World Cup for the first time in 60 years.
Insigne grabbed the crucial first goal after 56 minutes in the Group F clash in Naples, with Polish international Piotr Zielinski and Belgian Dries Mertens completing the scoring.
“We believed, we knew it was an important match,” Insigne told Mediaset Sport.
“We gave it everything and our sacrifices have been repaid. We believe in qualifying, we know it does not just depend on us, but what matters is to keep in the race and hope that City beat Shakhtar.
“I thank (coach Maurizio) Sarri for encouraging me, he has always trusted me. I will try to repay him with goals and assists.”
The Serie A leaders had lost 2-1 in Ukraine in the reverse fixture and needed to win to keep their slim chances of going through to the last 16 alive.
Manchester City have secured top spot with 15 points from five games with Shakhtar second with nine points and Napoli third on six.
Napoli now need Manchester City to beat Shakhtar and to beat bottom club Feyenoord in the final round of games to finish second on a superior head-to-head record.
The Ukrainians had the best of the first half at the Stadio San Paolo with the home side lacking inspiration and suspended Senegalese defender Kalidou Koulibaly.
Jose Callejon had an early chance but Andriy Pyatov saved on the line, with the Ukrainian goalkeeper also tipping a curling Insigne effort wide.
Napoli keeper Pepe Reina made key saves of his own, keeping out a driving Marlos strike and a long-range effort from Fred.
After the break Napoli broke through thanks to an inspired Insigne, with a trademark curling effort.
But Sarri’s side left their best football for late on, with two quick-fire efforts from Zielinski and Mertens sealing a victory to the relief of the paltry 15,000 home crowd.
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