After drawing blank in the opening two games, the Charity Stars FC striker was on a double on Thursday as the Black Starlets handed the hosts a 4-0 thrashing.
The win lifted Ghana to the top of the table, securing a ticket to the round of 16.
"The first and the second matches were tough and I knew we would meet such challenges at the World Cup but I knew I needed to prove my clout," Ayiah said.
"I had chances to score in those two games but I didn't utilize them, so my coach [Samuel Paa Kwasi Fabin] taught me how to be composed in front of goal.
"I heeded to his advice and it helped me perform greatly [against India]."
Ghana will find out their opponents for the next round on Friday.
At the African championship in Gabon earlier this year, Ayiah scored four times to steer Ghana to the final where they succumbed to Mali.
The 17-year-old's four goals were enough to earn him the silver boot for finishing as the tournament's second highest scorer.