The Republic's hopes of making the Euro 2008 finals ended with a disappointing draw at Croke Park on Wednesday night.
Steve Finnan volleys home the equaliser for the Republic
Robbie Keane saw a header saved before the impressive Cypriots wasted three good chances in the first half.
Cyprus captain Ioannis Okkas hit the woodwork just after the break and Keane headed wide when well placed.
Stelios Okkarides headed Cyprus ahead on 80 minutes and Steve Finnan levelled with a fierce injury-time volley before Marios Ilia was sent off.
There were 15,000 empty seats in the Dublin stadium as the Republic set out to gain revenge for the embarrassing 5-2 defeat by the Cypriots in Nicosia last year.
Only a win would do for Steve Staunton's men and they started out in positive fashion with Paul McShane just failing to get the vital touch to Kevin Doyle's shot across goal.
The Cypriots grew in confidence after they repelled the Republic's early assault and they gradually took a grip in midfield.
But it was the Republic who created the first good chance on 29 minutes with Keane's header from a Finnan cross easily saved by Antonis Georgallides.
Yiasoumis Yiasoumi wasted two glorious opportunites three minutes later, the first when he was put clear but the striker mishit his shot and Shay Given collected.
Yiasoumi met a cross 40 seconds later but his header drifted just wide of the woodwork.
The Republic had another escape a minute from the break when Cyprus found themselves in a three against one situation but Marios Nicolau opted not to pass and instead he sliced an attempted lob to the relief of a stranded Given.
Republic boss Steve Staunton made two much needed half-time changes in midfield with Liam Miller and Aiden McGeady replacing Joey O'Brien and Andy Keogh.
Cyprus were guilty of poor finishing again in the 49th minute when captain Okkas was put clear on goal but his flicked shot past the advancing Given struck the outside of a post.
Despite playing poorly the Republic still looked capable of scoring and Stephen Hunt flicked a header just wide on 55 minutes.
Daryl Murphy came on for Hunt before Keane squandered his second good chance after 76 minutes.
Doyle provided a pinpoint cross for Keane but the Tottenham forward headed wide from eight yards.
The miss proved costly because four minutes later Okkarides found space to power a header past Given and give Cyprus a deserved lead.
The Republic pressed for the equaliser and it came in the second minute of added time when Finnan latched onto a loose ball and smashed home.
Cyprus midfielder Ilia was red-carded during the visitors' protestations after the goal.
Rep of Ireland: Given, Finnan, Kilbane, McShane, O'Shea, Joey O'Brien, Reid, Keogh, Kevin Doyle, Keane, Hunt.
Cyprus: Georgallides, Satsias, Okkarides, Chrisou Paraskevas, Charalampidis, Marios Elia, Garpozis, Makridis, Nicolau, Okkas, Yiasoumis.
Referee: Mikko Vuorela (Finland)