The visitors took an early lead, as Charles Pickel finished well in the 18th minute after being set up by David Okereke.
However, quick goals from Juan Jesus and Giovanni Simeone saw Napoli turn the game around and lead 2-1 at the break.
Afena-Gyan replaced Okereke in the 65th minute and got the late equalizer to send the game into extra time.
The 19-year-old striker then scored the winning penalty as Cremonese ran out 5-4 winners in a shootout against leaders Setie A.
“Great team effort tonight. Congratulations to @USCremonese for qualifying to the next stage of the #CoppaItalia. If we believe, we can achieve it,” he later wrote on Twitter.
The young Ghanaian has now scored two goals this season, both coming in the Coppa Italia, since joining Cremonese.
Meanwhile, Cremonese will also now face Afena-Gyan’s former club AS Roma in the Coppa Italia quarter-finals.