The visitors took an early lead thanks to Charles Pickel’s brilliant finish in the 18th minute after being set up by David Okereke.
However, quick-fire goals from Juan Jesus and Giovanni Simeone turned the game around as Napoli took a 2-1 lead into the half-time break.
Afena-Gyan replaced Okereke in the 65th minute and grabbed a late equalizer to send the match into extra time.
The 19-year-old striker then scored the winning penalty as Cremonese ran out 5-4 winners in a shootout against the Seti A leaders.
“Amazing team effort tonight. Congratulations to @USCremonese for qualifying for the next stage of the #CoppaItalia . If we believe, we can achieve,” He later wrote on Twitter.
The Ghanaian youngster has now scored two goals in the season, both of which have come in the Coppa Italia since joining Cremonese.
Meanwhile, Cremonese will now face Afena-Gan’s former club AS Roma in the quarter-finals of the Coppa Italia.