"We are pretty ecstatic," Hay said, "it was a very, very big moment. It was only a few days back that we had a pretty tough result against France.
"For our boys to turn it around in the manner they have, a very good performance against Syria and winning today against Paraguay is a big result for our team."
Hay said the team battled back from conceding the equaliser late in the first half and stuck to their task in trying to play their way through and around Paraguay despite what was at stake and the clock winding down.
"When Paraguay scored the equaliser it was disappointing but when we came into the dressing room we emphasised to the boys they had 45 minutes to do something special and to qualify. The boys really bought into it and we definitely put that little bit of disappointment behind us.
"I think we showed a huge amount of patience. We didn't try and force things and try and go long. We kept the ball and created a couple of good chances and the finish from Lucas, I'm very, very proud of him."
The energy and commitment of the New Zealand side saw the team open the match on the front foot, causing issues for Paraguay from a succession of set pieces and it was by that method Hay's side took the lead in the 11th minute.
Probert's whipped corner was met at the near post by Ashworth who glanced his header into the far side netting to give New Zealand the lead.
But from there Paraguay went up a gear and began to string together more regular phases of possession but without regularly threatening Michael Woud in the New Zealand goal.
Woud was called into action twice in the opening 35 minutes, down well to save a shot after a New Zealand turnover before an excellent tip over from a goal bound Paraguay effort just after the half hour mark.
With halftime approaching Paraguay drew level in bizarre circumstances, hitting the cross bar and the post in a flurry of action after a free kick which also saw Woud make an excellent stop.
But the danger wasn't cleared and Namandu found space to fire in a shot from the angle which took a deflection through a crowd of players on its way into Woud's net.
The second half turned on the dismissal of Namandu for a second caution with the New Zealand side pushing forward against the 10 men of Paraguay, Rogerson enjoying the first good chance - seeing his low shot bound for the bottom right corner turned aside by goalkeeper Miguel Martinez just after the hour.
New Zealand continued to press for a winner and 10 minutes later, moments after joining the action from the bench, Sarpreet Singh teed up Probert with an excellent cut back from the byline but the wide man slipped at the vital moment without testing Martinez.
Singh himself was unlucky not to put New Zealand ahead soon after, seeing his shot from distance touched onto the crossbar by Martinez but there was still time for the Wellington Phoenix player to set up Imrie's winner with a delicious scooped pass which, when slammed home by the New Zealand striker, sent the Kiwis wild as they reached the round of 16 as they did in 2009 and 2011.
The Men's U-17s will now face Group B winners Brazil at the Estadio Sausalito in Vina del Mar on Thursday 29 October at 9am (NZ time).
Hay said the team will enjoy the next 24 hours before turning their attention to one of the world heavyweights in what is another step in the football journey for New Zealand's youngsters.
"It's going to be difficult but the opportunity to play against a side like Brazil it might be one of those things that happens once in a lifetime," Hay said.
"We'll celebrate tonight and rightly so and then get our feet on the ground and start looking ahead to Brazil. It's a big part of their learning curve as footballers. It's not the be all and end all, it's hopefully just another step in terms of them kicking on to the All Whites or being professional footballers one day."
