The legendary Italian boss was highly complimentary of his team as they qualify for a major tournament for the first time since World Cup 2002 with play-off win over Estonia
Ireland's 1-1 draw with Estonia in Dublin gave them a 5-1 win on aggregate and ensured they join the final 16 for Poland and Ukraine next year.
Stephen Ward fired Ireland ahead before Konstantin Vassiljev levelled for the vistors in the second half.
However, the damage had already been done on Friday following Ireland's 4-0 first-leg play-off win in Tallinn.
"Obviously we are all happy, the team and the supporters. It is a pity we didn't win because we wanted to give them a good a result but we played well away from home," Trapattoni is quoted as saying by RTE Sport.
"We continued this way without losing the game. It is a good signal for our attitude.
"This evening was another game, a tough game and we're happy we didn't lose.
"In football it is possible to concede a goal. In [the] first half, two or three time Robbie Keane could have scored. They were not easy chances but he could have score two.
"We showed the Irish people we built a fantastic team and we deserved to qualify. We have a good team with good players. We improved our qualities.
"After we lost against Russia we started with a new mentality and attitude. The players believed our system.
"The team improved and they believed what we asked from them. For us that is very satisfying."
Trapattoni's Ireland could be paired against Fabio Capello's in England when the draw for the finals is made in December.
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