We have a 10 week old puppy in the house so I can sympathize. It does get better in time with training.
First, very good that you have a crate. 
Next, help her to understand that you are there and that the crate isn't a lonely and scary place but a warm and cozy den.
Kathy's got some great advice, put your pup where she can learn that you are near, help ease her fear of being alone.
For me I try to place myself in the pups position. I've just been stripped from everything I know and dropped into a totally new environment. AHH! Now I don't even get to have a warm body to sleep next to! What's wrong? Why is it so lonely? Someone come let me know it's okay... Then I try to figure out how I can use positive reinforcement to communicate which behavior I want. It's hard because they will cry and cry, but be careful not to reward the crying. Similar to a baby, there's a point when the child has to learn to sleep in their own room. You have to balance letting her know it's okay with molding the behavior.
You might search for Kathy's web site. She has some great articles on crate training. Trust me, I've been there recently. 
And, yes, a Chihuahua can be very well crate trained and sleep through the night. My girl was great! She even traveled with me when I worked for a hotel chain and had to go to different cities, it was so much nicer to have her with me (I slept better because I wasn't alone).
My parents and gramma's Chi's are all crate trained. Now, I'll also say that once they were crate trained and fully trustworthy on the potty training they were able to move to sleeping on the bed with the humans.
In a way you can say they *earn* it by proving they are trained so they get to move to the next level. 
tabitha
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