In the league's first Friday night start, Arsene Wenger's side was trailing 3-2 with seven minutes remaining but Ramsey grabbed the equaliser and Giroud headed in an 85th-minute victor.
The Frenchman, who has been at the club since 1996, has been given a chance to turn things around despite calls for his head from disgruntled Arsenal fans waiting for a first league title under his stewardship since he led them to the crown in 2004.
After Aaron Ramsey leveled in the 83rd minute, Giroud sealed the three points within two minutes on an evening of thrilling attacks gifted by fragile defending.
"I don't know whether it was a lack of concentration or just a lack of game management in the last ten minutes, but you then concede again from a set play in terms of being overpowered. He has a big weight in the squad and it was a relief for me that he made a decision to stay".
"I prefer to be on the pitch", the 30-year-old Giroud said, "but as much as I can I try to help the team reach its target".
Wenger has always been desperate to keep Giroud because he offers something so different with his aerial presence but was also willing to let him leave if he felt that he needed a guarantee that he would start more regularly in this World Cup season.
"We were punished on every single mistake by a Leicester team that was very efficient going forward".
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