The Irish moved to the top of their group following Saturday night's European Championship qualifier at Richmond Park.
Having picked up just a point from their two previous encounters, victory was essential. The Republic's best chance in the first half fell to Ger O'Shea who hit the side netting following a cross from Margaret Saurin. The breakthrough came after 57 minutes when the Croatian goalkeeper fumbled a long-range strike from Ciara Grant.
In the 87th minute the game was made safe when O'Shea rounded the goalkeeper.
REPUBLIC OF IRELAND: Byrne, Deasley, Tracey, Conlon, Saurin, Sclanlon, Power, Grant, McCormack, O'Shea, Kissane. Sub: O'Reilly for Saurin (80 mins).
CROATIA: Djozo, Pirsa, Focic, Logarusic, Kozic, Kovac, Matuzic, Besker, Milas, Danjanovic, Koljenic, Subs: Franic for Matuzic (44 mins), Jaksic for Koljenic (46 mins), Nemet for Milas (81 mins).
Referee: M Patrascu (Romania).