Adam Rooney's goal early in the second half at Pittodrie proved decisive as the Dons overcame the stern resistance of Ian Murray's Championship side.
McInnes said: "Dumbarton have averaged two goals a game away from home this season. You can see the threat they've got - they've a good level of player and good organisation about them, but we thoroughly deserved the win.
"One or two injuries affected Dumbarton, but they kept going and had a great chance at the end through Paul McGinn.
"There was always a need for concentration and professionalism, but I don't think anyone could deny we deserved to win the game.
"We'd have liked it to be more comfortable, but we're in the draw and the players deserve huge credit for getting us into another semi-final."
McInnes also praised Rooney after he took his Dons tally to six goals in eight games.
He said: "I felt we weren't using Adam the right way in the first half, but he's the type of striker who won't settle for not scoring. He never let his head go down and kept getting into the right areas.
"We felt Adam in this team would be able to convert chances, and he's doing exactly that."
Sons boss Murray said: "The players were excellent. You could see how many boys had wee niggles and played on through it.
"We have to be delighted with them. We came up here hoping to do ourselves justice in terms of performing to a level we think they can play at, and they did that. They've exceeded expectations from the outside all season, but we've always said we have a good squad.
"Today was a good, good test for them and they stood up to it.
"It was a big chance at the end and if we'd sneaked a goal I think we could have seen it out for a replay."