Arsene Wenger had made seven changes with Thursday's Moscow decider in mind and though the visitors were under pressure again early on, Tadic cynically clattering Reiss Nelson when the youngster dropped a shoulder, the game's first chance fell their way as Arsenal switched off.
Southampton took an early lead when mistakes from several Arsenal defenders and Petr Cech saw the ball bounce through to Shane Long and the Ireland forward finished from close range.
However, substitute Charlie Austin pulled the Saints level with a tap-in in the 73rd minute.
Victory lifted Arsenal to within three points of fifth-placed Chelsea, who were held at home by West Ham, while Southampton remain 18th, three points away from safety.
Jack Stephens was sent off in stoppage time for the Saints after shoving Wilshere to the ground in an off the ball incident while Mohamed Elneny also got red carded moments later for an off-the-ball incident as the game turned sour.
Southampton's defeat kept them in the bottom three, three points behind Crystal Palace but with a game in hand.
"We had to dust ourselves down and come out with a real goal in the second half, and I thought we did that", he said.
The Gunners have a 4-1 advantage from the first leg.
Arsenal have won six matches in a row in all competitions for the first time since May 2017.
Arsenal edged a bitter five goal clash 3-2 against Southampton on Sunday.
Pierre-Emerick Aubameyang has been directly involved in seven goals in his first seven Premier League games (six goals, one assist), more than any other player in their first seven appearances for the Gunners.
At 2-2, he missed an open goal but he didn't let that knock his confidence as he scored the victor soon afterwards.
"I personally feel for what we have gone through and the way we have recovered, this team deserves huge credit because we were questioned really".
Speaking to Sky Sports after the game, Welbeck said: "It's a competitive game, Saints are in a hard position, same as us".
The game looked to be petering out before an on-field coming-together between Jack Wilshere and Jack Stephens led to the latter being sent off.