Sanchez broke the deadlock in the 60th minute of what had been a cagey game, selling the Saints defence a dummy before finishing left-footed for his 20th goal of the season, before Giroud headed a second with seven minutes remaining.
It was a fairly dismal evening for a Southampton side who had drawn their previous two games 0-0 and mustered few chances in another largely toothless display.
Arsenal moved above Manchester United into fifth on 66 points, three adrift of fourth-placed Manchester City with both having played 35 games. Arsene Wenger's side will also hope to reel in Liverpool, who have 70 points from 36.
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.
Yet Sanchez became an increasingly pivotal figure after the break and opened the scoring with a fine piece of individual skill before playing a role in the second goal, crossing for Aaron Ramsey, who looped it up for Giroud to head in.
It was a pleasing night for Arsenal manager Arsene Wenger, whose side have hit form at the right time, following Sunday's 2-0 victory over top-four rivals Manchester United.
"The whole team was dynamic, focused and showed a convincing desire to win the game," Arsenal manager Arsene Wenger told the BBC.
Saints boss Claude Puel, however, was left to ponder his side's lack of a cutting edge.
"It’s often the same against the big six," he said. "We cannot find a win. Every time we play good quality football with chances but without the clinical edge and it’s harsh on the players."
