A late goal from loan striker Callum Hendry was enough for the Dons, who had enjoyed plenty of chances in the first half but found Sons goalkeeper Sam Ramsbottom in inspired form.
The hosts had their own chances to win the game and were six minutes away from forcing extra time until Hendry netted the winner.
Speaking after the game Sheerin paid tribute to the efforts of Dumbarton, who currently sit second-bottom of League One.
"Dumbarton were really disciplined in their shape; they made it hard to get round them and forced us to play wide and then defended their box really well.
"From our point of view we could have been more ruthless in the first half with the chances we had but Sam Ramsbottom had two or three excellent saves but the fact we stuck at it was really pleasing.”
While Sheerin was glad to have won the game and progressed in the competition, Hendry’s goal was the just second Aberdeen have scored in 11 matches.
He continued: “It’s very much a relief [to have won]. The longer the game goes on, the more precarious it becomes.
“We spoke about staying patient before this game but obviously with the position we're in and the lack of goals in recent weeks and the longer that goes on the more desperate you become.
"But we stressed before the game that we do need to stay more patient. I thought second half we became more desperate than needed to be, throwing things in and allowing them to defend their box, which they did brilliantly.
"When you look at the number of chances we had, it was inevitable it was going to come at some stage.
"Stephen Glass arrived last night and he's now got a ten-day period of quarantine. The likelihood is he will miss St Johnstone then be in looking to hit the ground running."
Sons boss Jim Duffy hit out at his team's "unfair" schedule as they were dumped out the Scottish Cup following four games in eight days.
The League One part-timers finally ran out of gas seven minutes from time despite a spirited showing but manager Duffy could not hide his anger after seeing his team forced into a hectic run of games following the emergence of Scotland's lower leagues from lockdown last month.
He said: "It's not fair. It wouldn't be fair on any club to play four games in eight days.
"Would you ask that of top clubs? I absolutely doubt it. Today, you wouldn't have known it was our fourth game in eight days with what the players gave.
"I think sometimes we are a bit disrespectful to part-time players because they have to go to their work, keep themselves fit, and play a succession of games. I've got nothing but praise for my players."