Waghorn, who hit a hat-trick against holders Liverpool in the previous round, struck along with Jordan Anderson, who penned a pro deal last week, in the 2-0 win at The Valley.
Sunderland assistant academy manager Kevin Ball said: "It was a fantastic win and I'm really pleased for them. They've worked very hard and to come and get a win is a great result.
"Waggy's goal sparked us into life and then all of a sudden they got a goal back. The tempo was lifted and both teams contributed to a great game of football.
"If you want a chance like that at the end to fall at anyone's feet, it's Jordan Henderson. It was a great goal.
"Our back-four - and our keeper - were great, the two lads scored great goals but we won it as a team."