I have applied to a job and now have received credentials on a relevant skill / ability after having already applied (cover letter + resume + transcript). I informed the employer before hand that I had taken the exams (i.e. to show my seriousness about the work), but I did not receive the exam results until just now. Should I update the employer to tell them that I passed the exams? Or would this only be relevant if the employer decides to interview me?
If the exam is a strong requirement of the job - yes, inform them. You know your field better than we do.. but a "strong requirement" would be closer to something like passing the bar (as a lawyer), getting your bartender's permit, or having a driver's license for a delivery job.
If the exam is in the "desired" category of job requirements, not listed as a requirement, or you know that for most positions of this type, it's not a legal requirement or a huge limiting factor to being able to do the job... skip it. If you get a call back for an interview, mention it to the recruiter or the hiring manager.
That's true for many tech certifications - a CISSP, certification in a given language or system or other accreditation is usually NOT a must have. People are able to be similarly skilled without the course. It gets fuzzy with things where having a certification entitles the company to certain additional valuable things - for example, companies who are Oracle Partners keep a certain number of Oracle Certified DBAs and get a good deal on things within Oracle.
Only you will know how useful this specific certification is in this specific case. The line between annoying/clueless and a useful piece of information lies in being able to show that you can make the judgement call on how useful this is.
However, if you are dealing with an automated system or an external recruiter - do it. Don't question the usefulness - just do it. Automated systems don't care if you bug them, and it keeps your resume fresh. External recruiters like to know anything they can use to hype you up to the jobs they are advocating you for.