Interesting question. The short answer is: yes, exams were oral (and in Latin!), but they were really more arguments or discussions than quizzes. Universities were smaller and the students and teachers knew one another well, so ultimately it was the teachers' opinions of each student, not a transcript, that mattered.
For the longer answer, we can start with Cambridge's website:
The teaching took the form of reading and explaining texts; the examinations were oral disputations in which the candidates advanced a series of questions or theses which they disputed or argued with opponents a little senior to themselves, and finally with the masters who had taught them. Some of the masters, but by no means all, went on to advanced studies in divinity, canon and civil law, and, more rarely, medicine, which were taught and examined in the same way by those who had already passed through the course and become doctors.
So exams were mostly oral; undergrads would have to argue with grad students, and grad students would do the same with the doctors.
The situation in the early US appears to be similar. According to Lester Hunt's book:
Initially, as at Harvard in 1646, exams were oral and aimed to assess nothing more than the amount of information the students had packed into their minds. Later, using written tests, student writing and reasoning skills were assessed as well. Eventually, exam by recitation would dwindle and disappear, lingering mainly in the dreaded doctoral "orals." According to Mary Lovett Smallwood's history of grading in five early colleges...the first known grading system comparable to ours was instituted at Yale in 1785...Essentially, it was a 4-point system using words rather than letters and numbers."
So again, mostly oral exams. It's worth remembering that these were smaller classes where the students would have personal relationships with the teachers over multiple years. So, exams leading to formal written grades were probably less important than the subjective opinions that the teachers would form. This is similar to graduate study or even seminary today -- there may be some "objective" graded written tests, but the student's reputation with the teachers (e.g., through letters of recommendation) is far more important.
It's also worth noting that when evaluating students (up until ~1830 or so), the criteria were not only academic. Behavioral conduct (e.g., keeping one's room tidy), social conduct, and chapel attendance were also explicit metrics to be considered (some discussion of this is here).