The result leaves the Gunners three points behind fourth-placed Manchester City, and four points behind third-placed Liverpool with a game in hand.
With six minutes left, Arsenal made sure of the points when Sanchez and Ramsey combined to set up Giroud, who had the relatively simple task of heading home from close range.
Southampton are fighting Leicester City and West Bromwich Albion for eighth place in the Premier League and can take over from the latter if they beat Arsenal on Wednesday. "I can't see them finishing in the top four".
"To have all these chances and get into so many good positions is a good thing".
The former Saints youngster was enjoying a bright spell down the right but there were no further opportunities of note in the opening half-hour.
Arsenal goalkeeper Petr Cech was busy as he denied Manolo Gabbiadini from breaking the deadlock.
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Alex Oxlade-Chamberlain went off with a hamstring injury before half-time and was replaced by Hector Bellerin, a big blow for Arsenal considering his excellent performances in the last few months. The second bet is in respect to the number of goals being scored.
"It's not because we lack character if we lose games that people expected us to win, but this is a good opportunity to come back and recover".
With the goal, Sanchez also became just the fifth Arsenal player to reach 20 Premier League goals in a season.
Ian Wright, Thierry Henry, Emmanuel Adebayor and Robin van Persie are the only other Arsenal players to score 20 Premier League goals in a season, while the Chile forward has now scored more away from home this term - 14 - than any other top-flight player. And with very short technique he has invention in his head. He's creative, not phased by anything, and he surprised everyone in the stadium tonight with his goal. "Everyone was expecting him to take a shot, but he's very quick on the move".
"I feel sorry for him on that front, but we have more strikers available now because we had the situation where Giroud was not clear at the start of the season, we didn't know when Welbeck would come back, and in the end there was a congestion there".
"The whole team was dynamic, focused and showed a convincing desire to win the game", Arsenal manager Arsene Wenger told the BBC. But let's not go too quickly to conclusions. He's always a slow starter and gets stronger and stronger.
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