Trivino won the American League Reliever of the Month Award, and Hader won the National League Reliever of the Month Award after both pitchers were perfect in save opportunities during the month.
Trivino collected six saves for Oakland and posted a 0.71 ERA in 13 appearances, with 11 strikeouts in 12 2/3 innings.
The right-hander didn't allow a run until the final day of the month, putting together a streak of 12 consecutive scoreless appearances before then. His six saves were tied for second among AL relievers behind White Sox closer Liam Hendriks.
Hader picked up eight saves for Milwaukee and didn't allow a single run all month. He also racked up 18 strikeouts in 11 innings.
The left-hander's 11 scoreless appearances in June extended his scoreless streak to 17 games going back to May 22. Hader hasn't allowed a run in 31 of his 33 relief appearances this season, giving him a 0.55 ERA that's the best among relievers.