Mackie broke the deadlock at Pittodrie with his first goal for the Dons since February 2009 when he was on hand to tap the ball into an empty net after Miller had knocked the ball past keeper Marian Kello on the hour mark.
Miller added a second for the Dons when he tapped home Chris Maguire’s cross with 14 minutes to go.
The result ended Hearts’ five-match unbeaten run as the visitors never looked like troubling the Aberdeen defence.
Miller started for Aberdeen, with his current suspension only applying to league duty, while Derek Young was also handed a starting berth at right-back.
Maurice Ross returned to a makeshift Dons defence that was missing the suspended Richard Foster and Davide Grassi along with the injured Andrew Considine.
Hearts also made three changes, with Australian Rocky Visconte handed his debut at left-back and midfielder Ian Black and teenage striker Scott Robinson also starting. Striker Christian Nade dropped to the bench, while the suspended Ruben Palazuelos and the injured Lee Wallace both missed out.
Visconte made a confident start, sending a searching cross into the box that eventually fell to Gordon Smith but the youngster failed to test Jamie Langfield with a weak effort.
The visitors continued to surge forward and could have opened the scoring inside five minutes had Black managed to connect with Robinson’s inviting cross.
Darren Mackie should have broken the deadlock at the other end after 17 minutes when he stole the ball off Algerian defender Ismael Bouzid but, with Marian Kello stranded off his line, the Dons striker chipped the ball over the crossbar.
Marius Cinikas was handed the first booking of the match after fouling the advancing Mark Kerr on the edge of the Hearts penalty area after 26 minutes. Charlie Mulgrew tried his luck from the resulting set-piece but fired the ball wildly over the target.
Cinikas was fortunate to avoid a second booking following a needless foul on Aberdeen midfielder Paul Marshall only a few minutes later.
A fantastic save from Kello prevented Mackie netting with a powerful header after 35 minutes, with Fraser Fyvie failing to hit the target from the rebound.
Bouzid was next to rescue the visitors with a brave header just before the break to clear a Young centre with Mackie again lurking.
Some unsportsmanlike play from Aberdeen defender Ross angered the away support early into the second half. Eggert Jonsson had kicked the ball out of play to allow Dons midfielder Fraser Fyvie receive treatment for a head knock but instead of giving the ball back to the visitors the full-back threw the ball off Suso Santana before launching a Dons attack.
The Dons continued to take the game to their opponents with Ross crossing for Mackie but the Aberdeen striker again failed to convert after 57 minutes.
But the striker redeemed himself on the hour mark. Peter Pawlett released Miller with an intelligent pass and the striker clipped the ball past the onrushing Kello for Mackie to turn the ball into the unguarded net for his first goal of the season.
Hearts made a double change after 65 minutes with Nade and David Templeton coming on for the ineffective Smith and Santana.
Kello was again in inspired form to stop the home side doubling their advantage when the Slovakian tipped a glancing header from substitute Maguire round the post for a fruitless corner kick.
But Maguire, who had come on for the tiring Marshall, was instrumental in creating Aberdeen’s second when his back-post cross was slotted home by Miller after 76 minutes.
Langfield tipped a Visconte drive over the bar before the visitors introduced David Obua for Robinson as they searched for a lifeline.
Nade could have pulled a goal back for the visitors with a surging run and a venomous drive but Langfield tipped the ball over once again to ensure it would be Aberdeen who progressed to the next round.