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.
"That's why it's so important we stand up for the LGBTQ+ community," she added.
MTV stood by the location in support of the community as the awards show returned in-person after last year's ceremony was held virtually due to the pandemic.
The show also drew inspiration from the Danube, the city's famous river, by featuring a water theme with stages that appeared to be floating.
Sheeran began proceedings with his latest single Overpass Graffiti and later on performed his hit number Shivers - both from his chart-topping album Equals.
Korean pop superstars BTS matched last year's haul to win the most awards for the second year running by also securing the best K-pop and biggest fans categories.