The Gunners trailed 3-2 heading into the final 10 minutes at the Emirates Stadium, having earlier taken the lead inside two minutes through debutant Alexandre Lacazette.
Arsene Wenger's inspired changes paid dividends, though, as Ramsey levelled things up and then two minutes later Giroud turned the game completely on its head as Arsenal ended their opening-game hoodoo.
It is just the second time in seven attempts that the Londoners have picked up maximum points on matchday one, doing so in the most dramatic of circumstances.
Lacazette was off the mark in Arsenal colours with just his second touch of the ball - the first being from kickoff - as he nodded home 10 yards out from a first-time Mohamed Elneny cross.
It took all of 160 seconds for Leicester to get themselves back on level terms, with debutant Harry Maguire heading a deep Marc Albrighton cross back into the path of Shinji Okazaki, who was in the right place to turn the ball past Petr Cech.
Flying wing-back Sead Kolasinac and Welbeck saw another couple of chances come and go as the Gunners pressed for a second, but it was Leicester who found the game's next goal against the run of play 29 minutes in.
Arsenal never truly looked settled at the back with their now-familiar three-man backline and, after a mistake passing it out from the back, Albrighton pounced by sending in an inch-perfect cross for Jamie Vardy to convert in front of goal.
Groans could be heard around the Emirates soon after when Okazaki glanced a header wide, but those groans turned into cheers on the brink of half time as Welbeck rolled home after being picked out by Kolasinac alongside him.
A defence-splitting pass from Riyad Mahrez sent Vardy free early in the second half, only for Cech to charge out all of 35 yards to tackle the striker before he could truly compose himself.
Vardy had better luck soon after, though, this time climbing highest to pick out the far corner of the net with his head from a corner to double his personal tally for the night.
Oxlade-Chamberlain had another shot kept out by Schmeichel and Hector Bellerin was also denied from a one-on-one position, as Wenger's men began to turn the screw.
With 65 minutes on the clock Wenger turned to Ramsey and Giroud, the former almost making an instant impact by narrowly missing the target when getting on the end of a left-sided cross.
Leicester were having to hold on as they went in search of a rare win in this fixture, with Oxlade-Chamberlain on a one-man mission to find the net by sending his latest deflected shot inches wide.
The pressure eventually told as, in the space of two minutes late on, Ramsey controlled Granit Xhaka's pass and smartly tucked past Schmeichel, before fellow sub Giroud headed in off the bar from a corner to earn maximum points for his side.