It was goalless at the break, but second-half strikes from Alexis and Olivier Giroud proved to be enough to see off Claude Puel's Southampton.
Petr Cech was forced to make a couple of crucial saves to deny Manolo Gabbiadini and Nathan Redmond during the first half.
However, we held our nerve and took the lead when Alexis slotted home after leaving Maya Yoshida and Jack Stephens on the turf after a pass from Mesut Ozil.
The victory was wrapped up when Giroud nodded in from close range after Aaron Ramsey had headed Alexis' cross back across goal.
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