After two solid seasons back in the Premier League and a summer of spending, the Canaries' start to the campaign has fallen short of expectations.
An opening day draw with Everton at Carrow Road was last weekend followed by a 1-0 defeat at newly-promoted Hull, who were down to 10 men for the most part.
It is a disappointment that Norwich have responded well to and, having beaten Bury 6-3 in the Capital One Cup second round in midweek, they today overcame the Saints thanks to Redmond's second-half solo effort.
"It is a better day than last Saturday," Hughton said. "That's what the game is about.
"There was nobody more disappointed with the result last week, and perhaps the performance, than myself and the players.
"That being the case, you want to do something about it.
"It has been a good week and we got a lift off Tuesday night's game, winning and scoring some goals, and we were able to take that feeling a little bit into today.
"It was always going to be a tough game. Southampton are a very good side and they have invested very well on top of an already talented squad of players.
"I think in the end it is a good result for us, but I think over 90 minutes we deserved it.
"I think we shaded it. They had to go for it in the last bit of the game and it became very open, but they are a good side and I think they will have a good side."
The difference today was teenager Redmond, who proved a handful throughout.
The 19-year-old summer signing's pace often had Southampton on the back foot and saw him score the decisive goal in the 68th minute, cutting in from the left to unleash a fierce drive from the edge of the box.
"He is an exciting young talent," Hughton, who also managed Redmond at former club Birmingham, said.
"I always understand when somebody is in really good form, particularly if they're a young player, there will be interest in them.
"I think that is normal, but we have all been around long enough to know that also with young players they can have highs and lows and you have to be sure to look after them in the right way."
Redmond came close to several more goals today and Hughton also felt Norwich should have had a stoppage-time penalty after Robert Snodgrass was brought down by Jose Fonte.
The City boss did not, though, see the two first-half penalty shouts, with Calum Chambers handling after Bradley Johnson had done so more blatantly at the other end.
The latter decision by referee Howard Webb left Southampton manager Mauricio Pochettino dumbfounded.
"The overall feeling is that we played well and deserved a bit more from that match," the Argentinian said.
"We don't want to take credit away from Norwich, they actually played very well, but there was a penalty call that was not given and that would have changed completely the nature of the game. It was a penalty that should have been given.
"I don't understand why it wasn't given. The referee was very close to the action, so I really have no idea why it wasn't given.
"The referee was very close to the action, so I really have no idea why it wasn't given. It was blatant. It could have changed the game completely."
With two weeks until the next match due to the international break, the attention is now focused on the final few days of the transfer window.
Saints are hoping to shift a few players before Monday's deadline and there were reports that Jack Cork was subject of a bid from Fulham.
"I don't know anything about that," Pochettino said.
"There's no offer from Fulham, I don't know anything about what you're saying.
"He's a first-team squad member, he's a part of this team. If we start selling players we're only going to be left with just 11 players, so no."