The Red Devils will travel to Huish Park to take on the League Two club.
Yeovil are currently 21st on the League Two table and were the lowest ranked team for the fourth round draw after they defeated Bradford City 3-0 last Saturday.
"I don't think I could pick anyone better to go up against in the world we live in at the moment," Way said.
"It was an extraordinary achievement to get to the fourth round and to get Manchester United is a special moment for the club. The players and supporters deserve it. It's going to be a magical day."
It is the second time in four years the Glovers will take on United in the competition.