The Chilean, who scored 30 goals in all competitions last season, has declined to sign a new contract worth over £275,000-per-week with less than a year left on his current deal and has made no secret of his desire to play in the Champions League, a competition Arsenal will be absent from for the first time in two decades this season.
But Arsene Wenger's double change turned the game, FA Cup final hero Ramsey drilling in an 83rd-minute equaliser and Giroud notching the victor two minutes later with a header that hit the bar and bounced over the line.
Wenger said, 'We are open to an exceptional player.
Giroud, who had a hard time getting into the starting XI last season as Wenger preferred to play Alexis Sanchez as the lone striker, has been vocal in his desire to play more games from start.More news: U.S. warship challenges China's claims in South China Sea
Arsenal have been looking at paying Mesut Ozil £285,000-a-week to secure his future and they will now go to £300,000 for Sanchez.
His next touch was heading the ball into the back of the net in the second minute and the Premier League was off and running.
Within three minutes Leicester were level. Shinji Okazaki scored another header to make it 1-1 three minutes later, as the Japanese forward latched on to a Harry Maguire flick-on from Marc Albrighton's cross. It's no doubt going to spoil all of us because the rest of the games are going to suck by comparison. "We had some avoidable goals which we conceded and overall we have to work on that".
Danny Welbeck would bring the Gunners back level with a strike just before the half.More news: HRTC buses swept away in landslide, 50 feared killed
Jamie Vardy scored to put Leicester ahead just before the half-four mark after a cross from Marc Albrighton, while the bad Leicester centre-back (at least on this day) allowed Danny Welbeck to equalize.
Again a set-piece proved Arsenal's undoing as Mahrez's in-swinging corner from the right was nodded in by Vardy.
But the scoring wasn't complete, as two Arsenal substitutes inspired Arsenal's late fight-back.More news: Upton walk-off blast lifts Tigers over Twins
It seemed an unlikely platform for victory, but Ramsey neatly controlled Xhaka's dinked pass to drill in an equaliser and moments later Giroud's header from Xhaka's corner completed a remarkable turnaround.
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