Sharon Boyle's team dominated the game in spite of the Hungarian heat and a tricky opponent, but this victory sets them up well for Tuesday's clash with Bosnia & Herzegovina and Thursday's game against Hungary.
There were certainly plenty of positives to take from the opening performance, particularly the coolness displayed by goalscorers Carla McManus and Heather Payne.
The work that Boyle has been doing with this group was clear to see as they were confident in possession of the ball, swift in their movement, and constantly looking to push forward.
McManus, who came on as s substitute, opened the scoring on 76 minutes when she ghosted past two defenders to fire in. Then Payne sealed the win when reacting first to a free-kick from Aoife Lynagh to clip it into the back of the net.
"I thought that we were very good. We started off strong and dominated the first half with lots of chances on goal. We probably weren't as clinical in front of goal, but Slovakia were technically very good," said Boyle.
"We made a few changes, but the girls pushed on again and went at them. We got the first goal and really kicked on from there even though Erica Turner was excellent in goal for us.
"Slovakia had a couple of chances but we dealt with them quite well. I'm delighted with the win but this tournament is more about the development of the players and getting them minutes on the pitch. We will regroup after this, have a training session and go again on Tuesday with a few changes to our team.
"It's a great start for us and I'm delighted with the reaction of the players, which is the most important thing."
Republic of Ireland: Turner; Lambe (Fensom 70), Fennell (Grier h-t), Ruddy (Lynagh 76), Doyle, Toland, Lawlor (Cully 37), Baynes, O'Donoghue (McManus h-t), Mackey, Payne.
Slovakia: Gyomraiova; Papiernikova, Kratochvilova, Kalaberova, Srnkova, Patkova, Panakova, Dianiskova, Macugova, Karaffova, Ninajova.