Blues gifted Jed Wallace the opener after two minutes and failed to threaten the Millwall goal throughout until Ben Thompson's close-range effort 15 minutes from time sealed a third consecutive win for the Lions.
Karanka's side, one point from safety, now face a crunch match against fellow strugglers Sheffield Wednesday on Saturday before a visit from leaders Norwich next week.
"I don't need to look at the table because it is just a reflection of your job on the pitch," said the Spaniard. "Why am I going to look at the table if we are not winning games.
"We have to look at ourselves, our work and to keep believing because we have been really good this season. The only way I know is to keep working – to go to the training ground tomorrow and start preparing for the next game.
"One little mistake has cost us a lot tonight because of the position we are in at the moment, to concede a goal like that after two minutes is difficult to take.
"In a normal situation after that it would look like a team was going to go down but we were really good and we were in the game but we didn't defend well.
"Last week I said I wasn't happy with the performance but today I can't say anything bad because the players were fighting until the final whistle."
Millwall manager Gary Rowett was disappointed with his team's display despite picking up maximum points.
The Lions are now unbeaten in their last seven league games and moved into the top half of the table.
"I thought it was a poor performance in certain ways for most of the game," said Rowett.
"I think a marker was laid down in the first 30 seconds when the Birmingham central defender smashed it 100 yards back to us and it didn't get much better than that.
"I thought that when we got into the final third, Birmingham would be nervous. The goal comes from the defender under-hitting a backpass, Jed gambles and that is the sort of thing that happens to you when you're down there and haven't won many games.
"So it was a good start for us but I thought after that Birmingham were better than us. We didn't pass the ball with any fluidity and that meant we couldn't get out and attack like we wanted to.
"We've won four and drawn three now against some excellent sides, Norwich City and Watford included, and the players have shown what they are all about with a really good run of results."