Sanchez scored following a clever turn in the box and shot after 60 minutes, before substitute Olivier Giroud secured the points with a header as Arsenal kept their top-four hopes alive.
For a long time it looked as if Arsenal would be frustrated by a resolute Southampton defence, while Petr Cech was the busier goalkeeper in the first half, making good saves to deny Manolo Gabbiadini and Nathan Redmond.
Arsene Wenger insists he never questioned the character of his Arsenal side as they continued their turnaround with a vital win at Southampton.
"The whole team was dynamic, focused and showed a convincing desire to win the game". "We went through a bad patch".
Asked about Wenger's side finding their form at the right time, Mustafi replied: "Well I wish we found that before".
"All we can do is to keep winning our games and we will need another strong performance at Stoke on Saturday".
City will have played their home game with West Bromwich Albion by the time Arsenal kick off at Stoke, but a Gunners victory is sure to stoke up the pressure on Liverpool, who are now in third.
Mike D'Antoni to consider resting James Harden during next season
It's not an ideal matchup for him, but San Antonio doesn't have a chance without him playing ball something like he did in Game 6. He had four points on 1-for-6 shooting and four turnovers in the final five minutes of regulation and overtime.
Arsenal saw out victory with little trouble to complete a first league win at Southampton since their famous "Invincibles" season of 2003-04.
Ten days ago, Arsenal had dropped to seventh, eight points adrift of the Champions League positions - albeit having played one game fewer than fourth-placed Manchester City - after suffering a painful 2-0 North London derby defeat at Tottenham Hotspur. Maybe the Manchester City game gave us confidence.
"We had opportunities, they had one or two as well, so it was an open game but then in the second half we managed to come out and just keep playing our game, waiting and being patient until the opportunity came".
Southampton boss Claude Puel was frustrated after his side failed to take their chances with the game scoreless.
Wenger is likely to be without Alex Oxlade-Chamberlain at Stoke after he left the field in the first half with a hamstring problem. "We like to upset teams who come here and expect to win".
"He's got a small chance", Wenger said of Koscielny.
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