McCabe's strike midway through the first period proved decisive despite Wales creating several chances in the contest before and after the interval.
Wales won their opening match 2-0 against Hungary before drawing 0-0 with Czech Republic.
The Cyprus Cup finals day is on Wednesday, 8 March.
Wales Women's' Squad: Claire Skinner (Cyncoed), Laura O'Sullivan (Cardiff City Ladies), Emma Gibbon (Eastern Suburbs FC), Loren Dykes (Bristol City Women), Hannah Miles (Cardiff City Ladies), Hayley Ladd (Bristol City Women), Sophie Ingle (Liverpool), Gemma Evans (Cardiff City Ladies), Shaunna Jenkins (Cardiff City Ladies), Jess Fishlock (Seattle Reign FC - on loan at Melbourne City), Angharad James (Notts County Ladies), Charlie Estcourt (Reading FC Women), Bronwen Thomas (Brighton & Hove Albion Women), Georgia Evans (Bristol City Women), Nadia Lawrence (Yeovil Town Ladies), Helen Ward (Yeovil Town Ladies), Natasha Harding (Liverpool), Rachel Rowe (Reading FC Women), Kayleigh Green (Chieti), Melissa Fletcher (Reading FC Women), Rhiannon Roberts (Doncaster Rovers Belles), Amina Vine (Bristol City Women), Emma Beynon (Swansea City Ladies).
A first-half goal from Katie McCabe was enough to secure victory for Republic of Ireland in the final group game of the Cyprus Cup.