Arsene Wenger's side were seven minutes away from a miserable start to the new campaign after falling 3-2 behind at the Emirates Stadium.
Jamie Vardy then struck twice, either side of a Danny Welbeck equaliser, as the 2015-16 champions looked for a first league win over Arsenal since 1983.
But Arsenal's pressure in the final half hour turned the game back their way and Aaron Ramsey equalised with seven minutes remaining before Giroud headed the 85th minute victor.
Giroud then sent the previously despondent Arsenal fans wild when he headed the victor from a corner, Leicester keeper Kasper Schmeichel only managing to turn the ball onto the underside of the bar before it bounced over the line.
'I'm very pleased for the team tonight, hopefully we're going to carry on'.
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Tillerson's more moderate line, however, was seemingly dismissed by Deputy Assistant to the President Sebastian Gorka in an interview with BBC on Thursday.
"We kept the faith and kept believing in our game to push hard".
'If you look across the world of football this is a plum position, people want to be manager of Arsenal. For us we thought about the issue of who we wanted to take us forward.
"I'm happy the goals we conceded we can work on together".
Craig Shakespeare's side boasted numerous players, and characteristics, that saw them shock the world to win the title in 2015/16. They all deserve to play. "He's a fantastic guy as well, committed and loves the club". I opened the door for him because I know it's important for him to play but I don't want him to go.
More awful defending followed in the second half, and Vardy restored the lead after he was left wide open on a corner, with Petr Cech in no-man's land.
Cech admitted it was a sloppy display from the Gunners' rearguard, but he took heart with the way they recovered to snatch the points. "It was a relief for me he made a decision to stay".
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