Burnley struggled to recover from seeing their lead vanish in 11 minutes and a winner for the home side looked inevitable after James Ward-Prowse hammered a drive against the underside of the bar and Burnley keeper Nick Pope produced a magnificent reflex save to keep out a deflected Armstrong drive.
Southampton keeper Fraser Forster twice denied Wood in the latter stages - doing well to get a hand to a curling shot and then keeping out a close range header.
Burnley felt they should have had a penalty in the latter incident where Wood's shirt was pulled but Southampton deserved the points and should have had a fourth when substitute Che Adams had only Pope to beat but shot wide.
Southampton move up to 13th on 36 points, ten above 18th-placed Fulham. Burnley are on 33 points in 15th.
"It was a difficult start for us, and normally an opponent that is not helpful if you go two down," said Saints boss Ralph Hasenhuettl.
"Our decision-making in the final third was fantastic. We could have scored five or six or more goals against a team that doesn't normally give you many chances," he added.
Burnley defender Tarkowski was critical of his team's concentration levels.
"It's very unlike us, that. Our record of going ahead and getting a win or at least a draw is one of the best in the league, so to go 2-0 up, be very comfortable, it was mental sloppiness from everyone really," he said.