使い doesn't mean angel in all contexts, but does mean "messenger/bearer/...". In particular, the usual word for "angel" is [天使]{てんし} or [天]{てん}の使い = "messenger from heaven". Here you don't have 天, but 主 "Lord/God".
As snailplane points out in the comments, the phrase "Angel of the Lord" should appear frequently. Wikipedia writes:
The Angel of the Lord (or the Angel of God) is one of many terms in the Hebrew Bible (also: Old Testament) used for an angel. The Biblical name for angel, מלאך malak, which translates simply as "messenger," obtained the further signification of "angel" only through the addition of God's name, as ("angel of the Lord," or "angel of God", Zech. 12:8).
The same in Japanese. 使い translates simply as "messenger" and only obtains the further signification of "angel" only through the addition of God's name (or through the honorific prefix [御]{み}, as in ssb's answer). In that sense, the Japanese 主の使い seems to be faithful to the original Hebrew.