Hearts put in a great performance as they terrorised the Sunnybank defence in the first half and should have been out of sight by the break.
Confidence was shown around the park and Hearts never looked in danger.
Hearts started well and Kenny Mair went flying down the right and his cross found Ballard and he just missed at the far post.
Hearts kept up the pressure and Jordan Clark was unlucky when he screwed his shot just wide.
The pressure was rewarded when an Alex Davidson corner was headed home at the back post by Michael Bruce to put Hearts 1-0 up.
They kept up the tempo and another cross from Mair had Sunnybank wanting and Clark just missed at the far post.
Milne had a shot for Sunnybank by Fuerst was equal to the task.
Hearts should have doubled their advantage when Ballard was sent through but his left foot shot went wide.
But Hearts did go 2-0 up when a through ball found Clark on the run and his cross was hit home by Dane Ballard high into the net.
Fuerst then saw a header from Middler just go wide after a cross from Lunan.
Hearts were well worth their 2-0 half time lead.
Sunnybank came out trying to get a goal back and Milne came close with a shot that was just deflected behind.
The corner was cleared by a good Jamie Clark header.
Dane Ballard was sent through after an Alex Davidson pass and Clark saved well, then Davidson again got past the defence but his shot hit the side netting.
Sunnybank were awarded a penalty but the decision seemed harsh with few players even claiming a foul.
Lunan just beat Fuerst to make it 2-1.
But Hearts kept on pressing and were so unlucky when Richie Ligertwood hit the bar from the edge of the box.
Fuerst then saved a Page shot to keep Hearts ahead.
Hearts finally made the game safe when Dane Ballard beat the offside trap to cooly lob Clark with his left foot and put Hearts 3-1 ahead.
It was a result no more than they deserved.
Hearts’ next game is away to Newmachar United, ko 1.30pm.
Hearts - Fuerst, Park, Clark, Duncan, Keith, Bruce, Mair, Ligertwood, Ballard, Davidson, Clark;
Substitutes - Abolehassam, Thomson, Cunningham.