McLeish was unhappy with the way his side failed to deal with the long ball and set-piece that led to the goal.
McLeish said: "The start cost us the game, not the referee or the linesman.
"In the first minute, you want an inspirational tackle or header to get everybody going but that never happened. Instead, we didn't deal with a long ball and they forced a corner.
"We didn't learn our lesson because they score from the set-piece so it was fairly soft from our point of view.
"That made us nervous and, although there was fight in the second half, there was a lack of quality."
Coventry got off to a flying start against their midlands rivals when they took the lead after just 59 seconds.
Clinton Morrison forced a good stop from Birmingham keeper Maik Taylor before the resulting corner was side-footed into the bottom left corner by Coventry skipper Dann.
On 38 minutes Sebastian Larsson's inswinging free-kick took a deflection, forcing Keiren Westwood to pull off a fantastic save down to his left.
Coventry's new signing David Bell had a golden opportunity to extend the lead on the stroke of half-time but when Leon McKenzie's low cross found him in space his effort went wide of the left post.
The home side again started the livelier of the two teams before an end-to-end second half saw a good chance fall to Birmingham's Keith Fahey and Coventry's Leon Best within the space of a minute.
Larsson made up for his own error on 78 minutes when Best robbed the Swede in Coventry's half before playing Jordan Henderson in - but Larsson recovered the length of the pitch to block the shot.
There was controversy in the 81st minute when Sinclair had an effort ruled out for offside but Coventry composed themselves and saw out five minutes of injury time to seal the victory.
Coventry manager Chris Coleman said: "I thought we deserved it today. We created a lot of chances and could have won by more.
"Birmingham did have their moments but overall I thought we deserved the three points so I am delighted.
"There was a lot of enthusiasm in our performance and, to a man, I thought we were magnificent.
"As for promotion, our aim this season is to finish in the top half.
"Every team can beat every other team in this division but likewise you can be beaten by every team as well.
"But there are about 12 or 13 games left and if we go on a good run, who knows?"