Ibrahim Issah's 37th-minute goal gave home side Dreams all three points in Dawu.
Despite the loss, Hearts are still through to the next round of the competition and as Zone B winners, having held a four-point lead on the table coming into the game.
The Phobians are set to take on arch-rivals Kotoko, who finished second in Zone A behind Ashanti Gold, in the semi-finals.
Ashgold's opponents, however, will only be determined on Sunday, with second-placed Karela United and third-positioned Liberty Professionals the two teams in the running. They are currently separated by two points.
Karela play at home to Ebusua Dwarfs in Anyinase while Liberty host Wafa at Carl Reindorf Park.
In the last game of the zone, Elimina Sharks will host Inter Allies at the Nduom Sports Stadium.