The Phobians, led by interim manager David Ocloo, failed to impress as the Malians grabbed a deserved first-leg advantage with a 3-0 victory at Stade du 26 Mars
AS Real Bamako took the lead after 11 minutes, added a second in the 30th minute, and sealed the victory in the 75th minute.
Hearts of Oak's attack failed to put the Malians to the test.
Real Bamako are now clear favourites to win the tie and advance to the final qualifying round.
The second leg is set to take place on October 16 at the Accra Sports Stadium.