The Latin word used in the Vulgate (ca. 4th Century) was "Domini". For example, Matthew 1:20 in the NIV reads,

But after he had considered this, an angel of the Lord appeared to him in a dream and said, “Joseph son of David, do not be afraid to take Mary home as your wife, because what is conceived in her is from the Holy Spirit.

Here's the same passage from the Vulgate:

Hæc autem eo cogitante, ecce angelus Domini apparuit in somnis ei, dicens: Joseph, fili David, noli timere accipere Mariam conjugem tuam: quod enim in ea natum est, de Spiritu Sancto est.

The same word is preserved in our use of the term "AD" to designate years: it stands for "anno domini" which is generally translated "year of our lord". For example the US Constitition ends with a note that it was "done in the Year of our Lord one thousand seven hundred and Eighty seven".

In ancient Greek, the passage (Mt 1:20) was (one version, anyway):

ταῦτα δὲ αὐτοῦ ἐνθυμηθέντος ἰδοὺ ἄγγελος κυρίου κατ’ ὄναρ ἐφάνη αὐτῷ λέγων, Ἰωσὴφ υἱὸς Δαυίδ, μὴ φοβηθῇς παραλαβεῖν Μαριὰμ τὴν γυναῖκά σου, τὸ γὰρ ἐν αὐτῇ γεννηθὲν ἐκ πνεύματός ἐστιν ἁγίου·

It's less apparent than in Latin, but the word used is "κυρίου".

Isn't its usage rather anachronistic?

The words used in Latin and Greek that have been translated as "Lord" were not titles reserved for God, but they were general words which had roughly the same meaning as the English word "lord". I think the use of "Lord" in English is an appropriate translation.