The Sons had held Hibernian to a goalless draw at Easter Road last week, but their Edinburgh resistance was broken by first-half strikes from Osman Sow and Prince Buaben.
Jason Holt and Calum Paterson added further goals for the Jambos after the break, with Garry Fleming scoring a consolation goal for Dumbarton before Billy King’s late fifth for the hosts.
Hearts took command from the start and Sow almost gave them an early lead when he was inches away from connecting with Paterson’s cross.
A deflected shot from King was well-saved by Danny Rogers and Sam Nicholson fired narrowly wide before Sow had an effort blocked as the hosts turned the screw.
The visitors gradually began to settle but they went a goal behind in the 26th minute when Sow netted with a bullet header from Paterson’s right-wing centre.
Dumbarton hit back with Colin Nish seeing a shot blocked by Alim Ozturk before his point-blank header was superbly saved by Neil Alexander in the home goal.
But Buaben made it 2-0 from the penalty spot in the 40th minute, although his spot-kick, which was awarded after King was fouled by Scott Linton, was initially saved by Rogers before bouncing back over the keeper and into the net.
Hearts had further chances before half-time with Sow and Soufian El Hassnaoui both going close, while substitute Holt dragged a shot wide shortly after the restart.
Holt did find the net in the 56th minute when he surged through the middle of the Dumbarton defence before firing a left-footed shot into the left corner.
Fleming gave the visitors some hope with a close-range volley in the 73rd minute but Paterson sealed the win for Hearts when he headed home in the 84th minute from a corner from King, who wrapped up a fifth successive win for Hearts in the 88th minute.