Sort of what Josh wrote in his comment, it seems to have to do with the fact that the usual accent for this word would be on the ב, and it's moved to the ש because of nasog achor.1
Mishpetei Halashon Ha-Ivrit explains it as follows:
Basically, this nasog achor would require (based on the rule of dechik2) that the first letter of the following word have a dagesh.3 The meseg, then, partially restores the accent to its usual place, thereby removing the dechik.
(It does seem to be consistent for all cases where the root שבע is mil'eil because of nasog achor. Other examples include I Sam. 28:10, II Sam. 19:23, and I Kings 2:8.)
Nasog achor: the movement of word stress to a preceding syllable, to prevent two stressed syllables from being adjacent. (In this example, if the accent were in its usual place, you'd have "vayishaVA LO.")
Dechik: when one word ends with an unstressed open syllable (ending in a vowel sound) and the next one's accent is on its first (or only) syllable. The unstressed syllable is sort of "squeezed" (the literal meaning of dechik) between the two stresses, and this causes gemination of the onset of the next syllable.
Dagesh: A dot in a letter, which can change its sound from fricative to plosive (e.g., ב vs. בּ), or geminate (double) its sound.