Republic of Ireland midfielder Holland, a £750,000 signing from Ipswich in the summer, headed the opening goal from a Claus Jensen corner in the 11th minute to score his first goal for Charlton.
The 29-year-old's second came 14 minutes after the break when a clearance by Dugarry following another Jensen corner bounced off Holland's chest and into the back of the net.
Holland said: "In the second half Birmingham put us under a lot of pressure and Dean made two or three great saves to win us the game. It's a difficult place to come - but we deserved the victory."
Holland admitted he knew little about his second goal but joked: "You've got to be in there to score!"
Birmingham manager Steve Bruce said: "You can see why Charlton pick up points away from home, they are neat and tidy and always a threat.
"But I don't think we deserved to get beaten - I thought a draw was maybe a fair result because we played some good stuff at times.
"We had not conceded from a corner in 18 months but today we've conceded two.
"To be fair to Charlton they were good deliveries - but we are disappointed."
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