There are really three approaches I have used professionally to get someone to do what I need who isn't my subordinate (and sometimes then too). Here I am speaking of people I needed to work with in order to complete my tasks. The three things are finesse (I call it shmoozing), assertiveness, and currency.
Essentially finesse or shmoozing is simply getting someone else to like you enough to want to help you, and make your needs a priority over other people asking for things. I would do this in lots of ways and it's more of an art than science since you have to know what will work with that person. I am trained to do massage therapy. I would often offer to give someone a 10 min chair massage in their office while they looked over my proposal, discussed this or that with me, etc. I shouldn't have to do that to get what I need done, but it was highly effective and it got me a lot of increase in return time on requests. For others I might bring them in cookies here and there or for someone else a gift certificate for a pedicure. I considered these "the cost of doing business", and by being able to do more work in shorter time, I was also able to increase my income, such that it more than compensated for any money I spent shmoozing.
Assertive is pretty self explanatory. You make sure you aren't being overly passive and accepting less than the person is capable of, while maintaining you haven't gone so far that it's now feeling aggressive and hostile to the person you wish to work with more effectively. People respect assertive. They don't respect passive and they might even attempt to undermine aggressive as a way to pay you back if they feel you have stepped on their toes.
And currency. That is somewhat like finesse, but to me the difference is it should be more professionally aimed. Does Andrea need Susan to get her the reports faster? You can work to get in better with Susan so that she gets the reports in faster to Andrea. You can then let Andrea know that you cashed in some favors to get her reports expedited. If you have influence on whether or not Andrea gets to work on project she wants, use that too and make sure to tell her that you are "putting in a good word for her about what she has done working with you". I also had a personal assistant and I didn't always actually have to have her. So if I wanted favor with someone, I may loan her out to help them catch up on something, or the like. It really will obviously vary based on the overall work environment, rules, etc and knowing where you have any wiggle room to be able to ethically gain favor with someone.
In the end too, it is always best if there is any way you can frame it for "Andrea" why doing whatever you need done is also good for her, not just you. For instance, I worked with engineers who wanted to be made project manager when we landed a job. So I might tell Andrea that I really need those numbers on materials ASAP because I think we have a great shot at landing this and I would love to recommend her as PM when we do, but the sooner we can get those numbers, the better chance we have of getting that job at all. It's tight as it is, and we need to leave enough time to negotiate and make any changes they may need before the close date on bids. Then if it was a good enough job, I'd give her a chair massage, a pedicure gift certificate, get Susan to hurry those reports, lend her my assistant, be assertive AND every other trick I had. ;)