The contest's only goal at St Andrew's came from Sean Maguire in the 23rd minute, his second goal in successive matches.
North End in the main held the Blues at arms' length after going in front and in the first half were the side with more attacking prowess.
In the second half they had to dig in a lot more but did so to claim their first three points on the road in the league this season.
Neil said: "I thought it was a deserved win to be honest.
"In the first half we dominated the game and any team who puts two banks of four behind the ball like did Birmingham did, are hard to break down.
"The pitch here is six yards narrower than ours, so that make it hard to break through them - they were very direct. We had to be patient and move the ball well.
"It panned out exactly how we expected it to and planed for. The second half Birmingham was a bit more aggressive because they were a goal down
"Getting the goal was a big turning point, when you come to these places you need to get the first goal .
"If you don't, they will sit back even more and absorb pressure so that was crucial.
"In the second half we got into some better positions than we did in the first half but were a little bit wasteful. That meant we couldn't put the game to bed.
"However, it was a really pleasing win at a difficult place and we can go home happy."
The winning goal came after a free-kick from Paul Gallagher hit the post and rebounded across the face of the goal for Maguire to find the net.
Neil said: "We know Gally has great quality, every time we have a set play we can cause a problem.
"Sean was in the right position at the right time and got his foot over the ball to guide it into the net.
"We deserved to be winning the game at that stage because we were trying to win it and they were set up not to lose it."