I can relate. I'm also an introvert and picking up or carrying conversations feels unnatural to me. My only advice to you, ask questions, odd & random question, preferably open ended, stay away from "you" questions such as:
how are you?
what do you do?
what have you been doing?
These type of questions are susceptible to one worded reposes; "good", "I work at ACME", "nothing"
Instead, whenever possible ask questions that force her to describe her answers with; "what", "when", "where", "why" & "who":
What's a day working at ACME like?
When/How did you meet JOHN?
Where did you end up?
How was that like?
Why did you do it?
Who where you with?
Try to ask question that cause her to describe her answers by describing her senses:
What did you see?
What did it sound like?
How did it taste?
Was it uncomfortable? / How did you feel?
The smell must of been awful?
The above is to aid in some chit chat, if you want to be fun; add sarcasm, exaggerate & tease her responses. Smile.
Essentially, you want her to give you mid to long responses, and use what you learn from these answers, so you can build a smart ass remark (think C&F), clever joke, sarcastic remarks or follow-up questions, jokingly make fun of her (tease her)
Smile.
Take this poorly constructed scenario for instance. By convenience, you met up with her and now your sitting with her somewhere you two can have a conversation without distractions (I'm thinking somewhere in a mall).
...
Y: what are you doing later today? (open)
H: nothing, just heading home, i'm tired
Y: (confused face) its only 9'oclock, I didn't know you had a bed time (sly smile, tease)
H: (smile) shut up, i'm tired from walking
Y: walking where?/why where you walking? (open question)
H: I walked around the mall looking for a gift for aunt
Y: What did you get her? (close question)
H: A belt
Y: Why did you get her a belt? (open)
Y: You must not like her very much (tease)
H: Nooo, It's Gucci & it's really pretty!
Y: Ok let me see it (interaction*)
H: She takes it out.
Y: (sly smile) ugh it's hideous! (tease)
H: (smiles) no it's not!
Interaction is good. making her move, play little games, etc.
Make sure your teases are obvious, smile and change your tone to convey sarcasm, I personally had a problem with this in the past and my "teases" came out as insults
.Smile.