Both sides had plenty of chances as Steven McGarry struck the upright and Travis Dodd was magnificently denied by Heart ‘keeper Clint Bolton, while Fred wasted Heart’s best.
But Mebrahtu, a 77th minute substitute, broke the deadlock in the 89th minute giving Heart their first win in four games.
The Perth Glory started brightly with possession, but the Heart continuously gave the ball away allowing the Glory to control proceedings.
Despite Glory’s dominance, the Heart were very dangerous as Josip Tadic struck the side netting, before Nick Kalmar’s long range thunderbolt was punched wide by Perth custodian Danny Vukovic.
The Red and White were ready to pounce on the counter attack and that’s exactly what happened on 39 minutes as Matt Thompson found David in space down the left, but the forward’s weak effort was hit straight at Vukovic.
Just before the break Perth’s best chances involved Shane Smeltz. The former Gold Coast striker was found in the box by a superb Dodd cross, but the New Zealander directed his header wide, before the 31-year-old dragged his next effort wide.
On the edge of half-time ex-Heart defender Dean Heffernan was lucky to be still on the pitch after it seemed the fullback struck Tadic. Jacob Burns demanded that the Croatian got up, which saw Heart Coach John Aloisi confront the Perth Captain down the tunnel.
Heart started off brightly with David Williams looking for the decisive opener from distance, but Vukovic solid in goals for the Glory.
Just after the hour Melbourne Heart custodian Bolton denied a Dodd poke from point blank range with an amazing one-handed diving save to keep the tie level.
The Glory went even closer shortly afterwards however McGarry’s curling free-kick from the left clipped the upright.
Both sides were up for the contest and continued to push in search of the winner and Fred should have found it, although his diving header from Williams’ excellent cross was somehow directed wide.
But win one minute left on the clock Tadic played a nice through ball which Mebrahtu ran onto before poking through Vukovic for his first A-League goal.