A first-half goal from Jon McKain and a series of match-winning saves from captain and goalkeeper Eugene Galekovic secured Adelaide's progress to the final eight of the ACL for the second time in five years.
Galekovic thwarted the visitors with a series of near-impossible stops, including a sensational diving one-handed effort to deny Nagoya and Australia international target man Josh Kennedy.
McKain's effort to force a loose ball in from point-blank range in the 40th minute after Reds defender Iain Fyfe's header from a set piece led to a goalmouth scramble was enough for the home team to progress.
The win set up a two-legged quarter-final on September 19 and October 2 or 3 with a yet-to-be announced opponent.
Trani said the feeling following the win was "indescribable" and would be a moment to savour for a long time, as the Reds get set for a five-week break before returning to prepare for the business end of the ACL and the 2012-13 A-League season.
"What this win means for the club is indescribable really. It's something we are very proud of," Trani said.
"This particular tournament required high standards of performances, we have achieved that, got into the quarter-finals and can now enjoy this moment.
"We have some time from [now to] when we do play the quarter-finals to prepare, but it will give us the extra motivation through pre-season to able to continue on the success of the team and move forward for the forthcoming [A-League] season.
"Our players have become smarter and wiser for the experience in Asia, and we going to be a lot better for it."
Trani gave honourable mentions to goalscorer McKain and his defensive partner Nigel Boogaard, but reserved special praise for Galekovic.
"Eugene's performance was outstanding alongside our two centre-back defenders in Nigel Boogaard and Jon McKain through their ability to read the situation," Trani said.
"But Eugene obviously gave a match-winning performance."
Nagoya coach Dragan Stojkovic also paid testament to Galekovic's "match-winning performance", but bemoaned his side's inability to find the back of the net.
"Galekovic was absolutely fantastic today, the most valuable player and hero of the game - his saves were incredible," Stojkovic said.
"We tried really hard to play good football and to get a result, but obviously we were not good enough up front.
"We didn't use our chances and paid the price for not scoring goals, and the now the ACL is finished for us.
"We must find a solution as we turn our focus to the J.League."