McInnes had made three changes to the side that defeated St Johnstone last week, with captain Russell Anderson restored to the starting line-up in place of on-loan Alan Tate.
Barry Robson and Cammy Smith also started, with Niall McGinn on the bench after playing for Northern Ireland in midweek and Peter Pawlett dropping out of the squad.
Championship side Dumbarton had lost 3-1 to Queen of the South last week, and recalled goalkeeper Stephen Grindlay in place of Jamie Ewings. Defender Michael Miller stepped in in place of Mark McLaughlin.
The Dons' first chance came five minutes in as Shaleum Logan made the best of Robson's short pass to send a teasing cross along the six-yard line. Andy Graham cleared for a corner, but a little too close for comfort to his own goal.
Once the nerves of Ian Murray's visitors began to settle, they started to come into the game. Left-back Scott Linton showed their attacking intent as he burst into the Dons' box, but pulled his shot well wide of the target.
But, just after the quarter-hour, Linton was lucky not to find the net at the wrong end.
Johnny Hayes created the opening with a fine cross from the left that was headed towards goal by Robson. Grindlay got down to save, but the ball cannoned back off the full-back forcing the goalkeeper to grasp the ball before it crossed the line.
Dumbarton looked lively on the counter and, from a right-wing free-kick, former Don Mitch Megginson found space at the back post, but was denied by Jamie Langfield.
As half-time approached, Hayes swung in another cross from the left that the Dons' defence could only clear as far as Smith. He brought the ball down, but his left-footed effort was deflected away by Graham.
The sides were unchanged for the second half, but there was a moment of relief for Dons keeper Langfield as he failed to collect Mark Gilhaney's clipped cross from the right, his blushes spared by the presence of Anderson.
Scott Agnew twice threatened as the Dons continued their nervy start to the second half. His first effort was blocked by the challenge of Andrew Considine, before his second forced a save from the goalkeeper.
It was, though, the home side who opened the scoring after 53 minutes. Robson swung a corner to the near post, and Rooney got in front of his marker to head past Grindlay.
Boosted by the goal, Aberdeen had their chances to extend their lead. Hayes headed across the face of goal following a Willo Flood cross, and midway through the second half, Jack cracked a post from the edge of the area.
Dumbarton had used all their substitutes in the 10 minutes after the hour mark, with Colin Nish, followed off the park by Linton and Agnew. Garry Fleming, Scott Smith and Bryan Prunty took their respective places.
The hosts made their first alteration at the same time as their former striker, Prunty, came on - McGinn taking the place of Smith. Goalscorer Rooney was given a standing ovation when he made way for Scott Vernon after 84 minutes.
The final change saw Nicky Low replace Robson two minutes later, as the Dons looked to see the game out.
But there was a late chance for the visitors to snatch a leveller, as substitute Fleming's cross was met by Paul McGinn. The full-back did everything right, heading into the ground, but the bounce took his effort over the crossbar.
And there was one final opportunity for the Dons to add a second in stoppage time when Ryan Jack forced his way into the area, only to see his low shot saved by Grindlay.