Giroud tapped in midway through the first half to give Arsene Wenger's side a deserved victory over their north London rivals in a contest that they dominated for large periods.
Arsenal made two changes to the side that beat Fulham 3-1 last week, Laurent Koscielny and Jack Wilshere coming in for Bacary Sagna and the injured Lukas Podolski, while new signing Mathieu Flamini – in his second spell with the club – was named on the bench.
Erik Lamela was among the Tottenham substitutes, although fellow new additions Vlad Chiriches and Christian Eriksen were not included in the squad.
Arsenal seized the early momentum almost immediately and twice came close to going ahead in the opening five minutes. Santi Cazorla forced Hugo Lloris into a fine save with a curling free-kick before shooting narrowly wide from another set-piece shortly after.
The hosts' dominance looked to have come to an end as Tottenham began to dictate the game and press forward.
However, Andre Villas-Boas' men were undone on the counter-attack after 23 minutes, Giroud meeting a low near-post cross from Theo Walcott to prod home from point-blank range.
Arsenal wasted no time in attempting to add to their lead, Walcott again causing problems as he burst free down the right wing only to be denied by another athletic save from Lloris.
Aaron Ramsey fired over from distance on the half-hour mark before Lloris had to be fleet of foot to race out of goal and win the ball after Walcott had again broken through the Tottenham backline.
Villas-Boas will have been disappointed by his team's performance at both ends in the first half, with Andros Townsend the only player to force Wojciech Szczesny into a save with a tame effort.
Flamini's comeback came earlier than expected for Arsenal, the Frenchman thrown into the action two minutes before the interval after Jack Wilshere was forced off through injury.
Arsenal's superiority waned somewhat after the break, although Spurs still struggled to create any meaningful chances, Nacer Chadli sending a free-kick way over the bar in the early stages of the second period.
Tottenham's lack of creativity was nearly punished again by Giroud, the France international denied by compatriot Lloris, who produced a superb finger-tip save to deflect the striker's left-footed shot wide of the post.
Lamela was introduced midway through the second half, the former Roma man brought on for Townsend, while Brazilian midfielder Sandro was also thrown into the fray after Etienne Capoue was taken off on a stretcher with an ankle injury.
Spurs had their best chance of the game moments later as Szczesny kept out a vicious volley from substitute Jermain Defoe from the edge of the area, with Roberto Soldado's follow up blocked by Giroud.
Giroud almost turned provider as Arsenal launched another excellent counter, the 26-year-old pulling back for Walcott, only for Lloris to keep the deficit at one goal with another fine stop.
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