The race for second place in Group C took a painful turn for the Republic when the Swedes fought back to win 2-1 at the Aviva Stadium and put themselves into pole position.
There were boos from a capacity crowd in Dublin on the final whistle, but asked if that reaction signalled that it was time for him to go, the 74-year-old Italian came out fighting.
Trapattoni said: "No. I think we have done until now, a very, very great job, a great job - not a good job, a great job.
"We have changed many players in the squad and in the team and before the game, we were level in the table with a great team like Sweden.
"Our job until now has been very, very great. Obviously, supporters when they lose in Italy, here, in England, they boo. It's normal."
With Ireland, Sweden and Austria starting the night locked together on 11 points, it was the Swedes who were the big movers as Germany beat Austria 3-0 in Munich.
The Republic head for Vienna on Tuesday night, when Sweden play in Kazakhstan, knowing anything less than victory would wreck their feint hopes of a dramatic recovery.
Trapattoni, who had targetted four points from the double-header, said: "Obviously, our position at this moment is difficult, but our dream must stay alive because we have another very important game against Austria.
"Obviously, we need to beat Austria. It's not easy because Austria also lost against Germany, but until the table says we are completely out, we must think [we can do it].
"I must continue because there are another three games and in three games, anything can happen.
"We have an opportunity to continue this and we have to believe, also the team and dream.
"In my past career, I lost the Champions League final and one year after, we won the league and the Champions League. That's football."
It might have been so different for Trapattoni's men after they started the game well and took a deserved 22nd-minute lead courtesy of Robbie Keane's 60th senior international goal.
But they held their advantage for just 11 minutes, Johan Elmander getting ahead of Richard Dunne to head Mikael Lustig's cross firmly past keeper David Forde.
But as Zlatan Ibrahimovic dropped deep and started to pull the strings, James McCarthy and Glenn Whelan were unable to wrest back control in the middle of the field and the Swedes went from strength to strength.
The decisive moment arrived 12 minutes into the second half when 37-year-old Anders Svensson ran on to Ibrahimovic's defence-splitting pass to mark his record-equalling 143rd cap with the winning goal.
Sweden coach Erik Hamren was delighted with the resilience his team showed, but as he prepared for a tough trip to Eastern Europe, he was taking nothing for granted.
Asked if the quest for second place was now a two-horse race, he said: "We had a really important victory today, so it's a better situation for us, of course, with three points.
"But we have to win the games remaining. We are happy today and we have 24 hours to enjoy and be proud of ourselves, and then we will focus on the next game because as you saw with Ireland when they played in Kazakhstan, it's not that easy.
"They took a point from Austria too, so it's not an easy game. We expect a tough time there."