Sanchez opened the scoring on the hour at St Mary's before substitute Olivier Giroud made the game safe with a header seven minutes from time.
Victory took Arsene Wenger's side above Manchester United into fifth place in the table and left them three points below Manchester City, who occupy the fourth and final Champions League qualifying berth.
"We were focused and I felt that when we suffered, we stuck together," Arsenal manager Wenger told the BBC. "We have another clean sheet and I know we can go forward and score goals. The whole team was dynamic, focused and showed a convincing desire to win the game."
Arsenal must now win their three remaining games and hope City or third-place Liverpool drop points if they are to secure a 20th successive season of Champions League football.
Having previously taken two points from a possible 24 in away games against current top-half teams, victory was a welcome tonic. However, it came at the cost of an injury to Alex Oxlade-Chamberlain, who has found form in a new wing-back role.
"We don't know how bad it is," said Wenger. "It's a tight hamstring. We'll know more in the next 48 hours."
Arsenal went into the game on the back of a 2-0 win over Manchester United and procured the first sight of goal, with Aaron Ramsey firing wide after a foray by Oxlade-Chamberlain.
Saints had an opportunity to make the breakthrough when Arsenal's back three contrived to play both Dusan Tadic and Manolo Gabbiadini onside, but the latter's effort was repelled by visiting goalkeeper Petr Cech.
Oxlade-Chamberlain gave way in the 36th minute, with Hector Bellerin taking his place.