The Irish go into Tuesday's home game with Austria level on points with their visitors and fourth in Group C following Friday's goalless draw in Sweden.
The last time Trapattoni sent out a side for a competitive game in Dublin they were hammered 6-1 by current group leaders Germany.
But on the eve of Austria's visit the Italian said: "We have shown we are a good team, compact with good personality. The players played with conviction and good attitude in Sweden and are growing in belief that we can qualify.
"We must try win this game, but we must also play with intelligence because there are many games to come before qualification is finished. We have to be in contention until the very last game.
"It will be important to stay up with Sweden because Germany are too far ahead. But I have a lot of trust in this group and I am convinced we can improve even more."
Qualfication for successive major tournaments has not been achieved since Jack Charlton's day but that is the goal for Trapattoni, who took the Irish to Euro 2012 in the summer.
Seamus Coleman missed out on the squad then but has started the past three qualifiers and will again play at right-back on Tuesday.
He said: "One of my aims is to get to a major tournament with my country after missing out in the summer at the Euros.
"I'm sure there are a lot of lads out there who didn't go to the Euros who want to play in a major competition, so there is good hunger in the squad.
"We are going into the game to win and I think we have the calibre of players to do so, and in front of our home fans there's no better place to do it. The game is there for the taking, really."