Following a 1-1 draw in the first leg of the clash, Jonathan Walters scored twice at the Aviva Stadium to seal a 3-1 aggregate win to ensure Ireland's participation at a third European Championship in their history.
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Ireland continued to press forward and they were given the chance to take the lead midway through the first half when a penalty was given after Daryl Murphy's cross had been handled inside the box by Ervin Zukanovic
Walters stepped up to take the spot kick and he made no mistake with a composed strike that he rolled into the bottom corner, with Begovic diving the wrong way.
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The Irish continued to pose a danger at the other end of the pitch and Begovic saw his weak clearance closed down Wes Hoolahan, but the midfielder was penalised for a handball with the goal at his mercy.
Ireland took their lead into the break, but it was Bosnia who began the second half on the front foot as Medujanin fired in a low drive that was comfortably held by Randolph.
However, the visitors failed to create any clear-cut chances and they fell further behind in the tie in the 70th minute as Walters was picked out by Brady's cross at the back post, before firing a strike past Begovic and into the far corner.
However, the Irish would not be punished, as they held off a late surge from their opponents, which included Vedad Ibisevic crashing a volley against the bar, to see out the match and confirm their place among the finalists in France next summer.