The Firstborn??

[Kalimac]

I've wondered that too (also reading the Silmarillion for the first time). Though did Iluvatar give the Dwarves their minds or did they just develop that way? He tells Aule that "they would not shrink from you" if they did not have their own minds, but I don't remember him specifically breathing independent life into them, so to speak.

The only explanation I can think of is that the Dwarves were not, technically, finished when they were "put to sleep" - they hadn't learned their language yet and were still very unclear in their understanding. Maybe you could think of it as being like a pregnancy - the Dwarves were started before the Elves, but weren't "born" (released from their caves, given language and so forth) until after the Elves. And since date of birth is more important in the world than the date of conception, the Elves would still be the Firstborn.

Or of course Iluvatar could just have meant "The Elves will still be the Firstborn because I want it that way." Sort of along the lines of "This has been willed where what is willed must be."