The 16-year-old defender has been on Newcastle's radar for over a year after making his debut in the Checkatrade Trophy last year at first-team level.
Matheson caught the eye in the third-round encounter when scoring this equaliser in the 1-1 draw before Dale bowed out on penalties.
After the game, Matheson said: “Wow. What can I say really?
“Never did I think I'd be 16 years old scoring at Old Trafford - that's a dream come true for anyone.
"It's surreal. I'm lost for words. It was a special night for me, the team, Bryan Barry-Murphy [the Rochdale manager] and everyone. It was just unreal."