Here's Cicero, congratulating his friend Atticus on the birth of the latter's daughter (Ad Atticum 5.19):
Filiolam tuam tibi iam Romae iucundam esse gaudeo, eamque quam numquam vidi tamen et amo et amabilem esse certo scio. Etiam atque etiam vale.
I am glad that you now delight in your little daughter in Rome, and though I have never seen her, I still love her and know for certain she is lovely too. Farewell over and over.
For having become father of a boy, I like the phrase filiolo auctus (for a girl, it would of course be filiola auctus), used by Cicero when he announced the birth of his son Marcus (as discussed previously on this site; Ad Atticum 1.2), somewhat curtly:
Filiolo me auctum scito salva Terentia.
This is to let you know I was blessed with a baby boy, Terentia [Cicero's wife] is well.
Not a Roman, but I also stumbled across this letter from Gottfried Wilhelm Leibniz to Johann Bernoulli from 5 March 1697, writing (with a nice dominant participle):
Oblitus nuper de nata filiola Tibi gratulari, id nunc facio ex animo congaudens. Filium ni fallor jam habes, precor inde multa Tibi gaudia et diuturna.
Having lately forgotten to congratulate you on the birth of your daughter, I do so now, happy for you from my heart. If I am not mistaken, you already have a son, I wish you that they may bring you great and long-lasting joy.