The 26-year-old, who came in seventh place at the Olympics in Tokyo earlier in August, hit the longest 19.01 meters.
Portuguese Auriol Dongmo won the competition in 19.35 meters.
Roo's personal best, which also counts as a Swedish record, measures 19.34 meters.
The competitions in the Hungarian capital are part of the athletics' Continental Tour, but the ball and discus have been given Diamond League status as not all of these competitions have previously been possible due to the corona pandemic.
Simon Pettersson was the only Swede in the discus competition.
The 27-year-old, who won Olympic silver in Tokyo, finished in fourth place after throwing 64.26 meters the longest.
Britain's Lawrence Okoye won at 66.89.