Raheem Sterling had put Chelsea ahead with his third goal in two matches, but Lavia's terrific whipped strike opened his Premier League account and drew Southampton level.
Armstrong fired Southampton into the lead on the stroke of half-time and they held on to move level on points with eighth-placed Chelsea, and ahead on goals scored.
"I find it amazing that he says his team find it fun working hard," the ex-Everton star said. "No one finds it fun running around as much as his team did tonight!"
Mitrovic and a Lewis Dunk own goal put the Cottagers 2-0 up.
Alexis Mac Allister's penalty was not enough to extend an unbeaten start to the campaign for Brighton, who stay fourth on 10 points, two more than sixth-placed Fulham.
Wilfried Zaha returned from injury to open the scoring with a superb curled effort, but in a thrilling finale Wissa struck with two minutes remaining to earn Brentford their third draw of the season.
They sit ninth with six points, one more than 13th-placed Palace.
Anthony Gordon made it two goals in two matches to give Everton the lead at a raucous Elland Road, but Sinisterra's low shot means Everton's wait for a first away win since May 2021 goes on.
Leeds are fifth with eight points, five more than 15th-placed Everton.