Having trailed to Craig Eastmond's 16th-minute opener, Southampton battled back and secured victory courtesy of goals from Jake Sinclair and Harrison Reed either side of the interval.
The win – which was Saints' third consecutive victory at the Silverlake Stadium – takes Hunter's team up to third place in the Elite Group with just four games left to play.
If Southampton end the Elite Group stage in the top three, they will progress to a knock-out tournament to determine the overall winner of the U21 Premier League.
And Hunter says his team look good value to achieve that aim on the basis of last night's performance.
"We want to finish in the top three and then test ourselves in the knockout stage – and there is no reason why we can't do that," he told Saints Player
"Eastleigh is becoming a fortress so I am very, very pleased [with last night's win] particularly given the way we started the game – we were right on the back foot, so there was plenty of character shown by everyone.
"We keep working on things, particularly on game management. We were poor in terms of how we started the first 15-to-20 minutes and we conceded off a set-play, which is not our style so we didn't like that at all.
"We dug in and clawed our way back, so we got better and better as the half went on then Jake has got himself another goal.
"We talked about being composed individually, and he did that in scoring the goal – he was very composed and very clinical."
Reed's stunning strike three minutes after the break was his first at under-21 level having only made his maiden appearance for Hunter's team last Friday.
And the U21 Coach reserved special praise for how the likes of Reed, Dominic Gape, Omar Rowe and Matt Target have adapted to playing at a higher level in recent months.
He said: "We've said to the young players that they are being picked here on merit because of the things they've been doing in training and the way they've conducted themselves at under-18 level and since stepping up to under-21 level.
"The whole philosophy that we've got at the Club is one of pushing players forward while looking after the ones that are late developers. There is a pathway into the First Team which is clear for everybody to see."
Saints' next game sees them visit second-placed Manchester United next Tuesday, when they will be looking to beat the Red Devils at the fourth time of asking.
The two teams met in National Group 2 during the first half of the season, playing out a three-all draw in Altrincham last October before another tie at St Mary's two months later.
Both sides progressed to the Elite Group stage, and United triumphed at Staplewood back in February thanks to two goals in the final seven minutes after Cody Cropper had been sent off.
And Hunter is targeting revenge at Moss Lane next week: "We've got tough games coming up, but it's important that we stick to our guns and do the things that we want to do," he said.
"We've drawn with them twice and lost at home having being one-nil up, so we want a bit of vengeance.
"We'll train and travel on Monday then do some work on Tuesday morning to get ready for that game as professionally as we can do.
"We need to keep playing that we want to play and keep developing the players."