The Republic of Ireland must get the better of Estonia in a two-legged play-off in order to qualify for next summer's European Championship.
Tuesday's win over Armenia ensured Giovanni Trapattoni's men finished second in Group B and they were paired off with the Group C runners-up in Thursday's draw.
Estonia have never reached the finals of a major tournament, and this is Tarmo Ruutli's side's first play-off appearance.
Ireland have played in five-play offs before, winning just once, while their last appearance ended in acrimony when they were knocked out by France following a controversial handball by Thierry Henry ahead of the 2010 World Cup.
The first leg will take place in Estonia on either November 11 or 12, with the return leg at Dublin's Aviva Stadium on November 15.
"It is extremely positive for Ireland that we will play the first leg away," veteran coach Giovanni Trapattoni said.
"We should not underestimate our opposition, or any team, including Estonia.
"To have made it through to the play-off, they must have quality and mental fortitude.
"Our build up to this stage of 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."
Estonia finished 10 points behind group winners Italy in qualifying, with six of their points coming courtesy of wins over Northern Ireland.
The Republic emerged from a tough group containing winners Russia and the likes of Armenia and Slovakia to secure second spot by four points.
They have only reached the European Championship finals once before - in West Germany in 1988 - but will be relatively pleased to have drawn Estonia.
There were tougher-looking assignments in the pot of unseeded teams they could have been put up against, with Turkey hosting Croatia, the Czech Republic taking on Montenegro and Bosnia-Herzegovina going up against Portugal.
Former Republic of Ireland striker John Aldridge has backed the team to beat Estonia and qualify for the finals.
Asked if he thought Trapattoni's side would go through, he told Sky Sports News: ''Yes. We've got the greatest travelling fans and it would be great to see them on the road again.''
The former Liverpool player believes the Republic were fortunate to draw Estonia.
He said: "It was probably the easiest of the four (seeded teams to draw).
"We travel better than we play at home, we've had better results away than we have had at home."
Aldridge added on his hopes for the first leg: "They will try to get a clean sheet, try to get a win which would set us up nicely."
Aldridge admitted the suspension of striker Kevin Doyle, still waiting to hear the length of his punishment after being sent off in the qualifier against Armenia on Tuesday which the Republic won 2-1, would be a blow to the team's chances.
However, he backed Shane Long to fill the breach.
"Shane Long has done really well when he has deputised for either Robbie (Keane) or Kevin so it's a great opportunity for him.''
Ireland and Estonia have met twice before, in qualifying for the 2002 World Cup.
The Irish, then under Mick McCarthy's management, won both games 2-0, with defender Richard Dunne, a scorer against Armenia, on target in each match.
