A 63rd-minute penalty from Leigh Griffiths was the difference between the two sides, ensuring the Dark Blues ended a barren run of six defeats in a row.
McPake said: "First and foremost, it was about getting through to the next round and we achieved that. A lot of the players and myself have all been successful in football but have always had to come and battle it out at places such as Dumbarton.
"I wanted us to be more ruthless and we should have been out of sight as we had numerous chances, in particular from cross-balls, but we eventually got the goal through Leigh Griffiths and that's all that matters for us in terms of progression for the next round.
"Leigh Griffiths is a top player and somebody I've known since he was 16. His record speaks for itself and is still one of our best strikers when fit and firing. He won us the game with that penalty and he deserves a lot of credit as he looked hungry."
Dumbarton boss Stevie Farrell was proud of his players, who played the second half with 10 men after Ross MacLean was sent off.
He said: "It's always difficult when you get reduced to 10 men and you still have a long way to go in the match against a Premiership outfit and we still showed bravery despite being a man down.
"We had a go and stuck with two strikers to try and cause them problems. We weren't troubled at the back either, with Sam Ramsbottom only having a few saves to deal with. Over the piece we can be proud and we wish Dundee all the best in the next round."