At 3-2 up the Foxes looked set for a surprise win that would have sent shockwaves throughout the league but Aaron Ramsey's composed finished and Giroud's header meant Arsene Wenger started his 21st campaign in charge of Arsenal with a win.
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.
Everton are known to in the market for a new striker this summer, and following Arsenal's £52m purchase of Alexandre Lacazette, Giroud was heavily linked with a move to Goodison Park where he would be more likely to be the main striker.
Record signing Alexander Lacazette opened his account within two minutes of the kick-off, but the home side's lead was nearly immediately rubbed out as Shinji Okazaki equalised minutes later before Jamie Vardy put the visitors in front after half an hour.
Before that though, Vardy stunned Arsene Wenger's men with two goals and Shinji Okazaki also went on the score sheet to nearly punished Arsenal which played without star forward Alexis Sanchez because of injury.
Speaking in an interview with Sky Sports, Wenger said: "We are open to an exceptional player".
USA stocks snap 3-day losing streak in roller coaster week
Stocks saw some strength during trading on Friday, regaining ground following the sell-off seen in the previous session. While gold, a safe-haven favorite, pared some gains, it was last up 1.2 percent, at around its highest since mid-June.
But the lead did not last long and within three minutes Leicester were level, Shinji Okazaki nodding the ball into the back of the net in the fifth minute.
"It is a relief and I believe as well a reminder to the players that they have hurdles on the way and that we can get round it, like tonight so that is quite positive".
The impressive Vardy scored his second in the 53rd minute to put Leicester in front once more, curling Riyad Mahrez's corner into the bottom far corner amid poor Arsenal marking.
"Of course you can not afford to concede three goals at home".
Within ten minutes of the restart, Leicester took the lead again; again through Vardy who headed home firmly from a corner where no Arsenal player attacked the ball.
Leicester showed plenty of endeavour as they came close to causing an upset in the Premier League's first fixture.
"When the substitutes came on they had enough quality to make the difference".
Wenger added: "We know defensively we can do better but overall we produced a top-quality game. He is focused and happy in the dressing room because we won the game".