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.
Wenger revealed after Friday's win that Giroud, who has scored 97 goals in 231 appearances since joining from Montpellier in 2012, could have left the Gunners this summer.
Despite a Danny Wellbeck leveller on the stroke of halftime, the Emirates Stadium crowd was growing increasingly anxious and, when Vardy scored his second with 40 minutes to play, they must have feared the worst.
Wenger thankful Giroud turned down chance to leave
The visitors took the lead again 10 minutes into the second stanza as Vardy grabbed his brace with a cute header from close range. But Leicester, which so astonishingly won the league in 2016, couldn't hold on for its first win at Arsenal since 1973.
Giroud's Arsenal future had been subject to considerable doubt after five years, with the £46.5m ($60.4m) club-record signing of Alexandre Lacazette expected to restrict his starting opportunities even further.
"You know, I must say I opened the door for him at some stage because I know that it's important for him to play".
"It is fantastic", Wenger replied when asked about Giroud's willingness to fight for his place. In the end he made a decision to stay and it was one of the great days of the recent weeks for me.
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"Not only as a player, he is a fantastic player, b ut as well in the group - he has a big weight in the squad and it was a relief for me that he chose to stay".
"With the first corner, we were late".
It's only the second time in eight seasons that Arsene Wenger's side have secured three points in their opening fixture. I thought the ball would go out as it was flying over me and suddenly there was a player.
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She also said that she was regularly faced with sexist remarks, such as comments about her looks. Peterson, and he did a photoshoot with alt-right photographer Peter Duke .