The minhag is to daven ma'ariv before sefiras ha'omer based on the fact that ma'ariv is more tadir (it occurs more frequently) than sefira (see first Biur Halacha, O.C. 489, quoting Chok Ya'akov). Parenthetical aside: if you daven ma'ariv later than immediatly after tzeis hakochavim, I don't see any reason why you would not count sefira immediatly at tzeis (esp. since sefira ideally requires counting days that are "temimos", complete) rather than wait for after ma'ariv (there is a tshuvah by the Sheivet haLevi along these lines as well). Most people seem to not be careful about this and wait until after ma'ariv, no matter how late, to count.
Getting back to the main point, the gemara (Sukkah 56) has a debate whether the bracha of sukkah or the bracha of zman (she'hechiyanu) is said first when eating in a sukkah for the first time. We pasken like Rav that the bracha of sukkah is said first. The gemara explains that even though the bracha of zman is tadir because it is said more frequently, the bracha of sukkah comes first because it is the special mitzvas hayom.
Question: since the sefira count of each night is the special mitzvas hayom of that night (or at least the time period between Pesach and Shavuos), why should sefira not take precedence over the tadir mitzva of ma'ariv?