Aleksandar Kolarov scored the only goal at the Aviva Stadium after 54 minutes with a thundering strike from 15 yards that went in via the underside of the bar.
The visitors were reduced to ten men 22 minutes from time when Nikola Maksimovic saw red for a foul on Daryl Murphy but, despite their numerical advantage, Martin O’Neill’s side failed to find a route back into the game.
Reflecting on what was Ireland’s first competitive home loss in four years, Meyler said: “I think we deserved something, probably three points.
“They had two good chances, Randolph made a great save in the first half which you would expect him to make being a good goalie.
“We were unfortunate with the second one, Kolarov has stuck it in the back of the net but overall for the 90 minutes, I think we were the far better side. In football, you don’t always get you rewards and tonight we haven’t.
“We can’t have too many regrets. We poured our heart and soul into the performance and gave it everything.”
As a result of the defeat, Ireland dropped down to third in Group D – a point behind second-placed Wales and five adrift of leaders Serbia with two games still to play.
Next up for Meyler and co is a home game against Moldova before a potentially crucial final group stage fixture in Wales in October, with the Tigers’ midfielder confident of the getting the results needed to get the job done.
“We are hugely disappointed but we have been in this situation before, our aim is to win those two games and I believe we will,” he added.
“It’s not dead. We’ve got two games left. We’re still in a good position, we’ve got Moldova at home and Wales away.”