Wesley Hoedt's own goal gave Liverpool the lead, before Joel Matip and Mohamed Salah both netted to give the Reds an unsurmountable 3-0 advantage which they held on to.
Speaking to the official website, Henderson insisted the Reds could do better: “We can always improve, definitely.
“We're in a good moment – we've got a good rhythm, some good performances and good results – so of course we're confident but there's still plenty of room for improvement, certainly after today. But, overall, we're happy with the win.
“It's tough against anyone in the Premier League. Southampton are a good team and when you let them play they can hurt you. At times they found some gaps and created some half-chances.
“It's just about keeping going and keeping the performance level high. I thought the lads that came in did really well – that's a big positive for us.”
Xherdan Shaqiri was given his first start as Liverpool manager Jurgen Klopp made three changes to the side that beat PSG last week, including bringing in Joel Matip and Roberto Firmino for Joe Gomez and Daniel Sturridge.
With plenty of rotation expected with games to come in a short period - the Reds welcome Chelsea to Anfield in the Carabao Cup on Wednesday - Henderson lauded the importance of new faces carrying on where others have left off.
“We have been training really hard, even the lads that haven't been playing as much as they'd like,” he continued.
“They are just waiting for their chance and when they get onto the pitch they want to perform for the team and win games. That's what we've done again.
“We need to keep going, as I keep saying every week. There's another tough test in midweek in a different competition. We need to just keep going and keep improving.”
Henderson was also keen to praise Salah - who netted his first goal in four games amidst worries that he is dealing with added pressure to recreate last season's incredible levels.
“I'm delighted for him, of course. He wants to score goals so it's important for him,” added Henderson.
“But the most important thing is the win – I'm sure he'd tell you the same. It's good for him, good for the team and we move on.”