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United's assistant coach Luciano Trani said the feeling was indescribable and that the Reds will head into their extended. break on a high.
"We've got into the quarter-finals now and we can enjoy this moment," Trani said.
"We've got some time from now until the quarter-finals to prepare with the extra motivation through preseason that there will be something that can continue the success of this team."
Jon McKain scored the decisive goal in the 42nd minute while captain Eugene Galekovic pulled off a string of saves to deny the Reds' Japanese opposition.
McKain bundled the ball over the line from a Vidosic corner after the playmaker's lofted delivery was headed on by Iain Fyfe. The Nagoya defense was caught napping by the former Adelaide captain who knocked the ball into 'keeper Narazaki and into the net.
The Reds now have more than three months without a competitive fixture as the ACL breaks until mid-September, while the Hyundai A-League will resume a month later in October.
In front of over 9000 fans United played a more attacking style of play against Grampus compared to its previous ACL matches with all three substitutes used to refresh the attack rather than consolidate Adelaide's one goal lead.
Trani said it was part of the side's game plan to attack Nagoya as they could rely on a strong defensive base to build from.
"We realized before the game that when we do have the ball we're encouraged to really get at them," Trani said.
"We still have a base of where we want to start from and that's something that we've now got a belief through the team that if we maintain those disciplines and standards we already have a platform for our victories," he said.
United's defense proved the difference between the two sides as they stoically held off a Grampus side desperate for an equaliser. Trani singled out members of the Reds' defence as they helped put United through to the next round.
"Eugene's performance was outstanding, alongside defenders Nigel Boogaard and Jon McKain," Trani said.
"It was a match winning performance."
A fantastic break by the Reds in the sixth minute saw Dario Vidosic carve out a clear-cut chance for Iain Ramsay by opening up Nagoya with a cross-pitch ball over the defence. Ramsay's pace allowed him to leave Grampus defenders on his wake, however his final touch let him down as he fired over the bar.
Nagoya created their first chance of the game in the ninth minute after a neat one-two between Fujimoto and Tamada opened up United's defence, only for the cut-back to be cleared over the bar for a corner by Adelaide defender Iain Fyfe.
Captain Eugene Galekovic's heroics kept Adelaide in the game in the 34th minute by denying Socceroo Josh Kennedy with an incredible reflex save. Nagoya floated the ball into United's box and found their towering striker whose header appeared destined for the net, however Galekovic dived low to his left and managed to tip the ball out for a corner.
United breathed a sigh of relief in the 54th minute as Nigel Boogaard made a last ditch effort to deny Nagoya a clear shot on goal. Fujimoto threaded the needle to unlock Adelaide's defence, but as Kennedy turned for a one-on-one with Galekovic his path was blocked by Boogaard who was on-hand to force the ball clear.
Galekovic was again denied Nagoya in the 63rd minute as a Fujimoto free-kick was nodded towards goal by Masukawa. The 'keeper's positioning put him in the path of the shot and parried the ball away from danger to further frustrate the Japanese opponents.
Fabian Barbiero could've sealed United's victory in the 83rd minute after jinking past a Nagoya defender's challenge and going one-on-one with the 'keeper. With time to pick his spot either side of Narazaki the midfielder could only blaze high and wide to despair of the United faithful.
United began to dominate play in the latter stages as Kosmina looked to the bench for an injection of attacking football. Introducing van Dijk, Kostopolous and Kamara gave the Reds added flair up front, however hey were unable to double their lead.