Do universities typically cover publication fees associated with open access publications? I am an undergraduate student in a private Indian university. Last semester I worked on a research project with another undergrad student and under the mentorship of a PhD scholar.

We have submitted a paper to an Elsevier journal which charges a mandatory open-access fee of $500 on acceptance of the paper. In such cases, does the university typically cover these costs or are they borne by the authors of the paper? 

Which department of the university should I contact to ask for funding for the same?
Talk to your professor who will know better than us. It's possible your professor has a grant that also has an open access component, in which case that's where the money will come from. It's also possible your university's library has set aside some funds for open access. Here's an example page from the University of Hull. If your university has something similar, you might be able to get funding from there.

If you find you don't have funding, you can pay the OA fee out of your own pocket (probably not a good idea even if you are wealthy), decline OA (if it's a hybrid journal which publishes both OA and non-OA papers), ask for a waiver (Elsevier might grant this since you are from India, which is a developing country), and finally withdraw and submit elsewhere. If you take the last option, do so quickly so as not to waste the journal's time.