The other matches will see Turkey against Croatia while Czech Republic face Montenegro and Bosnia meet Portugal after the play-off draw which included eight of the nine runners-up from the group stages of the qualifying competition.
Sweden, who finished as the runners-up with the best record, reached the finals automatically with the nine group winners.
The play-offs will take place on November 11/12 and November 15 with the first named teams at home in the first leg.
The four winners will advance to next year's finals being staged in Poland and Ukraine.
Seeded Ireland, who controversially missed out on last year's World Cup finals after losing a play-off when Thierry Henry handled the ball in the build-up to France's winner, start as the strong favourites to progress this time.
They will be making their sixth play-off appearance in either the European Championships or World Cup.
Their only two previous matches against Estonia ended in 2-0 wins during the qualifiers for the 2002 World Cup in South Korea and Japan, the last time Ireland reached a major tournament.
Estonia, traditionally one of Europe's weaker nations, exceeded expectations by finishing second in Group C behind Italy after beating Northern Ireland home and away but they would cause a major upset if they beat Ireland over two legs.
Ireland's Italian coach Giovanni Trapattoni was keen to play down expectations against a team ranked 58th in the world.
"We should not underestimate our opposition, or any team, including Estonia. To have made it through to the play-offs, they must have quality and mental fortitude," he said.
"Our build-up to this stage in the competition has shown us the importance of organisation, attention to detail and self-belief. That will be vitally important to us as we prepare for these games in November."
Estonian FA President Aivar Pohlak said: "This is what we wanted, but there shouldn't be any illusions that Ireland will be an easy opponent. They come from a very strong qualifying group which, in addition to Russia, included Slovakia and Armenia."
Turkey v seeded Croatia is a repeat of their epic Euro 2008 quarter-final clash in Vienna, when Croatia went 1-0 up in the last minute of extra-time, only for the Turks to equalise with the last kick of the game.
The Croatians then missed three penalties in the shootout while Turkey scored theirs and advanced to the semi-finals.
Turkey assistant coach Oguz Cetin said: "Croatia is an opponent we know well. They are a very strong team, and have shown a great performance during the group stage. We have similar styles of football to Croatia."
Croatia coach Slaven Bilic was in charge for the Euro 2008 quarter-final defeat and the CNN Turk news channel suggested he would be more fired up to win the clash than Turkey coach Guus Hiddink who did not live through the experience.
Okan Buruk, one of Hiddink's aides, said he had spoken to Croatian officials, adding: "They are not pleased with the draw either, we share the same emotions. They would have rather picked another team, and so would we."
Bosnia v seeded Portugal will be a repeat of the 2010 World Cup play-offs, when the Portuguese won 2-0 on aggregate to go through to the finals in South Africa.
Portugal coach Paulo Bento expects another very tough fight, saying: "Bosnia were just minutes away from taking the first place in the group against France. They have some individual pla
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