The 20-year-old scored the Young Lions' second of the contest in the 21st minute with a run that started from his own half and ended by him drilling a finish into the far corner.
Elliott was handed his first start in the competition for Lee Carsley's side, who had already secured top spot in Group C, with Curtis Jones an unused substitute.
England will now take on Portugal in the quarter-finals on Sunday evening.
Elsewhere in the group, there was heartbreak for goalkeeper Vitezslav Jaros and Czechia after a late goal in their meeting with Israel denied them a place in the knockouts.
Israel's Omri Gandelman netted in the 82nd minute to clinch a 1-0 win and demote Czechia to third place.