Colin Bell's side will now have to wait to find out which side is the best runner-up from Group A and Group B for their final game on Wednesday, March 8th.
It was another solid performance from Ireland who have now kept three clean sheets at the Cyprus Cup and never looked like conceding in a structured and determined performance.
The match-winning moment came with just 20 minutes on the clock as centre-back Diane Caldwell picked out the run of Katie McCabe with a fantastic 60-yard pass and the Arsenal Ladies forward controlled with her left before smashing the ball high into the roof of the net.
Claire O'Riordan, who made her first international start for the Women's National Team, came off at the break for Megan Connolly to ensure a defensive second-half in difficult conditions in Paralimni.
Connolly could have scored a second half-way through the second-half when Stephanie Roche found her on the edge of the area but her effort was well saved as Ireland went on to defend the lead and win the game.
Head Colin Bell was elated with the performance who has had an unbeaten start to his managerial reign with two wins and one draw.
Bell said: "It was another fantastic performance where the players have given everything for the cause and fought until the final seconds to get the win we deserved.
"It was a fantastic goal. A fantastic ball from Diane (Caldwell) to pick out Katie McCabe and it was a fantastic finish, a goal worthy of winning any game.
"In the second-half, we had to keep things tight so we brought Megan (Connolly) in who was able to drop in and give Karen (Duggan) more of a chance to drop back and defend the way she did.
"Wales are difficult opponents, a really good team but they didn't have one real chance in the second-half which is testament to how well we played.
"We could have snatched a second at the end but I'm delighted for the team who are getting the rewards for their efforts."
Republic of Ireland: Byrne (GK), Perry, Caldwell, Quinn, Scott, Duggan, O'Sullivan, Roche, O'Gorman, McCabe (Littlejohn 85), O'Riordan (Connolly 46).
Subs: Hourihan (GK), McQuillan (GK), McCarthy, Berrill, McLaughlin, Murray, Shine, Kiernan.