The Gunners ran out 2-0 winners at St Mary's; Sanchez and Olivier Giroud getting the goals to move Arsenal back fifth in the Premier League table.
Arsenal must avoid complacency and carry the intensity they showed in Wednesday's win over Southampton into their three remaining games to achieve a top four finish in the Premier League, manager Arsene Wenger has said. "Very good technique in a short space". He's always a slow starter and gets stronger and stronger.
"I am very pleased because we had a consistent performance from the first to the last minute", said Wenger.
Now Arsenal must keep the pressure on the top four by showing the same focus and winning their three remaining matches, starting at Stoke City on Saturday.
Arsenal boss Arsene Wenger insists his side have consigned their wobbles to the pages of history and are intent on chasing down what had appeared an unlikely Champions League place after victory over Southampton.
Arsenal's first chance came with 15 minutes played as Alex Oxlade-Chamberlain found Aaron Ramsey but his shot was just wide.
"We have another clean sheet and I know we can go forward and score goals".
"We can not find the good clinical edge", said Puel.
Having beaten Manchester United 2-0 on Sunday, Arsenal have loftier ambitions than a top half finish.
After a boring first half, Sanchez produced a moment of magic to open the scoring when the Chilean wrong-footed two defenders in the box before calmly slotting home.
Under-fire Wenger has constantly reiterated his side can not put a foot wrong in their remaining games, and they were solid enough to take home victory from the south coast.
Sanchez struck his 20th league goal of the season to open the scoring and Wenger was full of praise for the Chilean - whose own future at the club remains up in the air with his contract expiring next year.
Wenger's opposite number Claude Puel lamented his side's inferior "clinical edge" to the Gunners after squandering the best chances in a tight first period.
Southampton had a late opportunity through Shane Long, who headed straight at Cech, but this fixture further highlighted that they must strengthen in attack during the summer as they made it three games without finding the back of the net.