Everton gave the ball away with a sloppy piece of play in midfield on 52 minutes to allow the home side to break forward and Armstrong drilled the ball low into the bottom corner.
The home side were quicker to the ball and won the physical battle against Frank Lampard's team, who were sparkling in their 3-0 victory against Leeds United last weekend, but reverted to the form that has seen them dragged towards the relegation scrap this campaign.
Pickford was by far the busier of the two goalkeepers, saving brilliantly from point-blank range to deny Livramento, who flicked James Ward-Prowse's low corner goal-wards, while Mohamed Elyounoussi also struck the crossbar.
Southampton remain in 10th place with 32 points from 25 matches, while Everton are in 16th with 22 points from 23 games. Everton's next match is at home to Manchester City on Feb. 26.