However, every translation I have seen of this verse - the mechon-mamre above, the Metsuda Tehilim, and Artscroll, whose Gemara translation I am also using - translates neshef as "dawn". Why?
First of all, what you say is not exactly true. It might be the case that every English translation that you have seen understands the word to mean morning - and most probably because it is preceded by the verb "to rise" - but there are Hebrew commentaries that understand it in line with this gemara. Both the Radak and the Metzudat Tziyon understand the word as referring to the night-time, though the latter bases his interpretation on Isaiah 21:4, and neither of them mentions Berakhot.
That said, Targum Yonatan translates it at שפרפרא, which means early morning, and if you're looking for a Talmudic basis for this interpretation (which, most probably, post-dates this targum), then you can see where this is the meaning given to the verse in Ketubot 111b, acc. to Rashi (s.v. קדמתי).
As for discrepancies in general between the Talmud and English translations of the Bible, consider the following two points:
• For a start, where the Talmud provides translations of words, those translations frequently differ from the word's actual meaning, and have little to do with its etymology. This particular point, in fact, was one of the bones of contention between Saadiah Gaon and the Karaites. Where the latter stressed the need to check cognate languages for similar words, Saadiah emphasised the role of the rabbinic literature in defining Hebrew usage. As it is, the methodology of the Karaites was closer to what linguists today would do in order to understand words that turn up very few times within the biblical literature;
• Secondly, there are different translational methodologies, and how an individual translation renders an individual word depends very much on the methodology adopted by its translator. If you were to consult the Artscroll Tanakh, you will find that their translation often differs widely from the literal meaning of the Hebrew verse. This is an example of a "dynamic equivalence" translation, which means that it aims at reproducing the sense of the passage, where that sense is dictated (in their case) by their hashkafa.
A "formal equivalence" translation, like the NRSV (or - to a lesser extent - like the JPS) will care less about conveying the overall sense, which might in some cases be communicated in a footnote, than a word-by-word transferral of the Hebrew into English.
I mention these two points because the question you asked is perhaps of broader scope than just the translation of נשף in this particular verse, but is likely to bother you subsequently in relation to other words in other verses as well.