Sheeran, who opened the ceremony on Sunday, won best artist and best song for Bad Habits.
BTS picked up four awards including best pop and best group.
Rapper Saweetie called for tolerance as she hosted the event in the Hungarian capital amid controversy over recent anti-LGBT legislation in the country.
The US star insisted it is "always important we respect each other", as she relaxed on a chaise longue wearing a bikini in a mock spa location - a reference to Budapest's spa culture.
