First-half goals from Lee McCulloch and Darren McGregor set Rangers up for another home win on the back of Monday's 8-1 hammering of Clyde in the Petrofac Training Cup.
And although it was a gutsy performance from Dumbarton, a second-half strike from Nicky Clark and an own goal from Chris Turner left Ian Murray's side on the wrong end of another crushing defeat - with only a Bilel Mohsni own goal for consolation.
McCoist was understandably delighted with his side after they recorded their first league win at Ibrox of the season after losing to Hearts at home in their opening league encounter.
He said: "I was really pleased as the football was of a good standard. It's also really important to get among the goals but if you get them from different areas of the team it makes you really hard to play against."
On rumours he is in the frame to succeed Tony Pulis at Selhurst Park, McCoist added: "There has been no contact at all with Crystal Palace. I've got a job to do here.
"Anyone who knows me will be aware that I am desperately keen to get the job here completed - the immediate one being to get out of this division and that's where I see myself hopefully short-term and longer-term."
It was an unhappy return to Ibrox for Dumbarton boss Murray.
Although he was happy with his players' commitment, the former Rangers defender was less than pleased with way his side conceded the goals.
He said: "I was disappointed with the goals we lost and if people are not defending properly we will lose a lot of goals and that's four we lost for the second time."
Murray's thoughts then turned to Dumbarton's League Cup tie against Hibernian on Tuesday.
He added: "We can't change the last few games but the players are looking ahead to tomorrow night as Easter Road is a fantastic place to go."
Both managers acknowledged Mohsni could have been sent off early in the second half for a trip on Scott Agnew on the edge of the box, while McCoist thought the referee Brian Colvin got in right in only showing the Rangers defender a yellow card.