Saints took the lead on seven minutes when Newcastle could not clear their lines after a fine Karl Darlow save and Adams was on hand to volley home Theo Walcott's cross.
Walcott flashed a shot fractionally wide before the break and Oriol Romeu cracked an effort against the crossbar after half-time as Southampton pushed for a second.
Armstrong made sure of the points on 82 minutes with a powerful shot before goalkeeper Alex McCarthy denied Joelinton with a sensational save inside the final five minutes.
Saints are now top on 16 points, level with Liverpool having played a match more. Newcastle stay 11th on 11 points.
King of the Match: Che Adams (SOU)
Southampton: 21 Nov v Wolves (A)
Newcastle: 21 Nov v Chelsea (H)