Presidential candidate abraham lincoln had just won election or presidential candidate, abraham lincoln, had just won election

Presidential candidate Abraham Lincoln had just won election

Explanation:

Commas = non-essential clause, a clause that can be removed without affecting the essential meaning (grammatical and logical) of the sentence.

If the sentence does not make sense in context without the name, then no commas are necessary as it with the case of this question, so no need for a comma before and after the name.