Samuel Saiz scored his sixth goal of the season in the first half and substitute Stuart Dallas eased frayed nerves by ramming home a low finish in the final minute.
Victory lifted Leeds top for the first time since the opening day of the 2004/5 season, while they last looked down on the second tier for more than 24 hours when Howard Wilkinson's side won the old Second Division title in 1990.
"It's just a situation at the moment," said Christiansen, who admitted his players showed signs of fatigue as they extended their winning league start to seven matches and nine in all competitions.
"If you tell me it's the last game and we're in that position I would be happy, proud and all that you can say about that.
"But right now, don't give me anything. It puts a bit more pressure on the team because everyone now wants to beat Leeds.
"The performance was not the best, but if you still take the three points you have to be proud. What we have done until now is a high level."
Birmingham will feel aggrieved to have got nothing from the game. Their performance belied their poor start to the season and had most in a crowd of 31,507 praying for the final whistle as Leeds held on to their slender lead in the closing stages.
"I'm very happy with the result and also the way we took these three points. It was a very difficult game," Christiansen added.
"Birmingham gave us these problems, but when you manage to take the three points even when you play not so good, or the same performance as in the last game, then you're probably more satisfied.
"They didn't play to their best. It was obvious. We didn't control the situation as we wanted, but this is because we had a very good opponent."
Christiansen confirmed captain Liam Cooper is expected to be fit for Saturday's trip to Millwall after being forced off with a knock in the second period.
Blues boss Harry Redknapp was convinced his side deserved at least a point after slipping to their fourth straight away defeat.
"We completely dominated the second half. One-way traffic wasn't it?" he said.
"We just couldn't get a break when we needed it and then obviously, we're pushing six forwards up the pitch to try and get a goal and you get done on the counter-attack.
"I've got a new team. I changed a team that wasn't good enough, but I didn't get them in for the start of the season.
"Five games had gone and suddenly I get new players in and it's a different team now.
"I've had a week to work with that team, but it's one that's got potential. There's some good players there.
"People need to be patient. They've got players here now who can take them on to another level."
Redknapp added he was waiting to learn the full extent of hamstring injuries sustained by Jota and substitute Che Adams.