The margin of the defeat flattered Newcastle who offered next to nothing as an attacking force.
Southampton's fifth league win of the season, having lost their opening two games, moved them to 16 points from eight games and above champions Liverpool on goal difference.
It's a massive reversal of fortunes for Southampton, who were beaten 9-0 at home by Leicester on Oct. 25, 2019, and now they're top of the league.
They will stay there until Sunday at least if Everton fail to beat visitors Manchester United on Saturday.
"Amazing what my team does at the moment," said coach Ralph Hasenhuttl, who has transformed Southampton in the past 12 months, after the match.
"We maybe could have scored a few more today but the rest was nearly perfect. Good game management and everyone was on the highest level. It's a bit scary what we're doing at the moment, but not so surprising as I see them working."
Saints were without leading scorer Danny Ings, who underwent knee surgery after being injured in last week's win over Aston Villa, but it did not blunt their attacking threat.
Southampton swarmed all over Newcastle from the start and it took them only seven minutes to open the scoring.
Adams forced a flying save from Karl Darlow but the visitors failed to clear the danger and when Miguel Almiron lost possession to Theo Walcott, the former England forward teed up Adams to crash an unstoppable volley into the net.
Walcott, on a season-long loan from Everton, was at the heart of Southampton's best moments and was inches away from doubling his side's lead moments before halftime when his curling effort flew just wide of the post.
It was a perfect night for the 31-year-old who was playing his first home game for Southampton since leaving the south coast club to join Arsenal as a teenager in 2006.
Southampton's relentless pressing game hounded Newcastle into errors and the hosts twice went close within a minute after half-time. Jan Bednarek's effort got cleared off the line and then Oriol Romeu's shot was touched onto the bar by Darlow.
Walcott could have had a penalty before Armstrong wrapped up the points with a low finish.
"It's a fantastic achievement and honour to be part of it, this is where it all started for me," Walcott told Sky Sports. "We work hard for each other and we are there because of our performances. It was a worry at 1-0 because we had dominated so much. But Stuart getting the goal was much deserved.
"I've got some great memories here but I want to make some new ones."
Newcastle are in 11th spot on 11 points.