How to say one has been accepted directly at a competitive examination?

I'm not aware of general terms for this, but for your specific example of being admitted without examination, one could use (informally) admitted sans exam or given a pass (equivalent to receiving a bye, but I think slightly more likely to occur) or admitted on merit or prequalified. In formal justifications one might well find phrases like that which Barrie suggested, e.g., admitted on the basis of academic record, but unless the process is objective and well-documented, the envious might say admitted because he's got pull, knew somebody, had connections, or had an in.

At least in the US, there's no such system. Everyone takes the same entrance exam (SAT or ACT) no matter which schools they apply to. Then they send the schools an admission packet with their transcript and test scores. The schools pick applicants with the best overall records.

To imply someone was hand-picked by a school for outstanding performance, they could have been offered a scholarship. To imply that the school relaxed their exam requirements, they could be a recruited athlete or a legacy admission.

There's no concise way of saying it in English. We would have to say something like on the basis of the candidate's academic record.

At least one university calls it score optional admission:

Score optional review provides an opportunity for applicants to be considered for admission without submitting or in disregard of standardized test scores. Admission remains a highly competitive process, and score optional candidates will each be considered on their own merits.

accepted on academic record in lieu of exam. accepted exempt from exam.

The problem with "in lieu of" is conciseness. You need to say something was used in lieu of something else. However, since the concept is not common in English, using academic record in the sentence is more explanatory.