Ireland will face Finland in their next Davis Cup match in April after the 3-2 win over Estonia at the weekend.
Finland beat Bulgaria in Sofia in their tie, coming from 1-0 down in the process.
Sam Barry was the Ireland hero as they eventually saw off Estonia 3-2 at David Lloyd Riverview.
Barry and James McGee saved two match points in a remarkable doubles game on Saturday before taking a 9-7 win in the final set.
McGee had the chance to seal the win for Ireland against Vladimir Ivanov but he lost 6-4 6-3 1-6 4-6 6-1 and the match was level.
However, Barry was a comfortable 6-2 6-1 6-3 victory over Marek Marksoo.
The second-round tie will take place in Riverview in April.