Giroud came big for the Gunners once again.
Arsenal record-signing Alexandre Lacazette launched his English league career in style by heading in the opener after 94 seconds.
Albrighton crossed for Jamie Vardy to put Leicester ahead after Xhaka had gifted the ball straight to him.
The impressive Vardy scored his second in the 53rd minute to put Leicester in front once more, curling Riyad Mahrez's corner into the bottom far corner amid poor Arsenal marking.
The 2016 PFA Player of the Season had always been linked with a move to the Gunners and looks set to leave the Foxes this term, however it seems that Arsene Wenger is not keen. "We were quite good a year ago on corners and overall I believe that with a bit of work we can get that out of the system", the French coach added.More news: Arsenal's new €53 million man enjoys dream start to Premier League life
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"There were hard moments when we were down and we managed to get out of that and it's great to see".
"But you could say that Ibrahimovic transferred an identical amount of goals from France to England". "At the moment, sitting here, I've got a dejected dressing room".
"All pre-season he's looked sharp and fit", said Shakespeare.
Leicester striker Jamie Vardy scored twice but he admitted the Foxes have plenty of work to do on the training ground to cut out their defensive errors.
"We've been courageous to the end and thank God we won the game".
Fredrik Ljungberg marked his debut with the final goal in a 3-0 win over Manchester United in September 1998, while Emmanuel Adebayor got off on the right foot away to Birmingham City in February 2006.