While the University Libraries have subscribed to many academic databases and the functionality can vary a bit, all have some way to save a citation directly from the resource record. One typical spot is on the side with other useful tools as seen on the right.
When I click the Cite tool, I am given the option below to Export to Biographic Management Software:
The Export Manager will display many options, choose the RIS Format (note that Zotero is listed to make the choice clearer).
Choose Save, and the first time you will see the box below asking a) Do you want to export to Zotero and b) do you want it to export automatically to Zotero in the future.
The article is then stored within Zotero with all the required citation elements as well as a permanent link should a user wish to revisit it.
To store content to the Web, simply click the document icon next to the Zotero Z:
It will give you the option to "Save to Zotero as a Webpage"