It's actually worked best for us the other way around. When she's exhausted, like in the middle of the night, once she wakes up it's an absolute bear to get her back to sleep.
When we made the drive from MA to DC this spring, it worked well for us to start the drive fairly early in the morning (around 5:00 AM or so), but do most of it during the day. First of all, *I'm* in a better mood, since I hate driving at night. Second, my wife could sit back there and keep her somewhat occupied. The key was stopping every hour, hour and a half and letting her run around. What would be a seven hour drive for two people becomes about ten with her, but she was only really upset for a very short period of time, when we got stuck in traffic on the beltway. Otherwise, it was a combination of her being happy and dozing.
It's just a matter of being patient and factoring in an extra 50% or so of drive time. Ergo, the twelve hour trip to Columbus is either eighteen hours, or one place ride.