Jason Donohue's side conceded an early goal inside the first 15 minutes when Andreas Savva's shot beat Aaron Maguire in the Irish goal.
On the half-hour mark, Ben Quinn scored for Ireland to give the side a well-deserved equaliser. Midfielder Jamie Mullins, who was a late replacement in the starting line-up for John Joe Power, played a perfect through ball for Quinn, who scored past the Cypriot goalkeeper to secure the draw.
Speaking after the game, Head Coach Donohue revealed that he was happy with how the side coped in the Cypriot heat.
"The lads have never played in those temperatures before. Going in at 1-1 at the break, we were happy with. We had all the chances in the second-half though and were unlucky not to win," said Donohue.
The two teams meet again on Thursday afternoon. Kick-off is at 3pm (1pm Irish time).
Cyprus: Demetris Stylianides, Sergios Fenerides, Konstantinos Venizelos, Nikolas Lamprou, Konstantinos Patichis, Giorgios Louka, Demitris Spyridakes, Andreas Athanasiou, Andreas Savva, Konstantinos Koromias, Kyriakos Chrisomiles.
Republic of Ireland: Aaron Maguire; John Ryan, Sean Grehan, Ben Curtis, Tommy Fogarty (capt); Glory Nzingo (Aaron O'Reilly 59), Luke McGlynn, Jamie Mullins (John Joe Power 70); Cian Kelly, Michael Leddy (Evan Ferguson 52), Ben Quinn (Edwin Agbaje 59).
Referee: Antoniou Menelaos (CYP).