The striker went close on a number of occasions and was involved in both goals as the Young Lions ran out 2-1 winners in Rakvere, Estonia.
It means after a draw against Croatia in their opener in Tallinn, England need a point from their final group match against France to progress to the last four.
That final fixture is back in Tallinn on Monday.
England went ahead in the sixth minute through Arsenal's Benik Afobe after Serbia goalkeeper Filip Pajovic spilled Conor Coady's cross under pressure from Harry.
England had the better of the opening exchanges as Harry fired just wide and was then then denied by Pajovic.
Pajovic kept out Harry's header at the start of the second half and he went for a spectacular lob from half-way on the hour, but the effort drifted over.
On 63 minutes, Harry linked up with Afobe to set-up the second goal for Birmingham's Nathan Redmond.
Serbia made it 2-1 on 70 minutes and Harry was replaced 10 minutes from time.