Martin O’Neill’s side found out on Sunday who stands in their way of a place at next summer’s Euros in France, while the draw also threw up a Scandanavian derby.
The remaining two ties will see Ukraine meet Slovenia and Hungary face Norway.
Martin O’Neill’s Ireland side missed out on automatic qualification after losing to Poland in the final game of the group stages.
Their hopes of making the finals of a second successive Euros, therefore, will rest on how they get on over two legs against Bosnia in November, with the first leg away from home.
Mehmed Bazdarevic’s side finished third in their group behind Belgium and Wales, but a win against Chris Coleman’s men in the penultimate game did help them seal their place in the play-offs.
Elsewhere, Zlatan Ibrahimovic’s hopes of taking part in Euro 2016, which is being held in the country where he plays his domestic football with Paris Saint-Germain, depends on how Sweden fare against Denmark.
After missing out on one of the top two places in the group, Ibrahimovic said the tournament without his representation is unimaginable.
“I don’t want to miss the European Championship, it’s unlikely and unthinkable for me to see this tournament ongoing without me,” he said.
“We didn’t go in the classical way, so we will find out another one to go. Whoever will be our opponent in the play-off, it will be tough.
“In this kind of games, each team give its 200%. It’s an all-in. But I’m sure we will qualify.”
Scandinavia’s third team in the play-offs, Norway, were paired with a Hungary side who came within minutes of qualifying automatically.
They were all set to go through as the best third-placed finishers, only to be usurped by a last-minute Turkey win.
Finally, Slovenia, who find themselves at this stage having finished behind England and Switzerland, will take on Urkraine for the final spot.
The first leg ties will be played between 12-14 November, with the return matches on 15, 16 and 17 November.
Ukraine v Slovenia
Sweden v Denmark
Bosnia-Herzegovina v Republic of Ireland
Hungary v Slovenia