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The Republic controlled the game from start to finish but found their path to goal impeded by late tackles from the Slovenian defence. Ireland were awarded a series of free-kicks in the first half as a result but their efforts couldn't breach the home side's defence.
Niamh Fahey could have earned a penalty when she was taken down in the box but the referee waved Irish appeals away.
Ireland manager Sue Ronan left five of her first-choice players out of the starting line-up, with more testing games against Scotland and Germany later in the week in mind.
She brought on two of those players at half-time, Aine O'Gorman and Diane Caldwell, to give the side more zest on the wings and in attack.
That change yielded instant results when, just two minutes after the restart, Irish senior squad member Aine O'Gorman steadied herself on the edge of the box before tucking a shot into the corner.
Niamh Fahey from Salthill Devon put the result beyond doubt with Ireland's second goal, scored from a similar position to the first strike, after 60 minutes.
“Slovenia put 11 players behind the ball and it made life very difficult for us but they got tired in the second half,” said manager Ronan.
“We knew Aine O'Gorman would cause them problems with her pace if we brought her on and she did the business.
“Our objective was to win the game and get three points on the board. We've done that and we move onto the game against Scotland on Thursday in good spirits.”
Germany beat Scotland 7-0 in the other group game played earlier Tuesday.
Republic of Ireland: Aine Genockey (UCD), Seana Cooke (Raheny Utd), Lisa Toohey (Aisling Annacotty), Eilis Nevin (Dundalk), Grace Murray (Dundalk), Aoife Herbert (Ballina Town), Niamh Fahey (Salthill Devon), Nicola Sinnott (UCD), Stephanie Roche (Stella Maris), Noelle Murray (St. Anthony's), Susan Donovan (Ballincollig).
Subs: Aine O'Gorman (Stella Maris) for Sinnott and Diane Caldwell for Donovan (Both half-time), Meabh De Burca (Salthill Devon) for Fahey (65 mins).