In what was an even contest, the home side were a danger on the counter-attack and it was a break from Theo Walcott that led to the only goal.
The team pressed forward in attempts to construct openings, but clear-cut chances were at a premium.
Andre Villas-Boas opted for an unchanged team from last weekend's victory over Swansea City, with new boy Erik Lamela taking his place on the bench.
Hugo Lloris was extended in the first few minutes when Etienne Capoue was adjudged to have felled Tomas Rosicky, leaving an inviting free-kick chance for Santi Cazorla. Lloris tempted the Spaniard by leaving him a fair bit of goal to aim at, which was accepted, but the goalkeeper swooped across his line and parried away.
Cazorla was afforded another invitation moments later when a decision went against Mousa Dembele after a coming together with Theo Walcott. From a central position, the midfieler banked on the wall jumping and shot low and thankfully just wide.
There followed a soothing sustained spell of possession that unsettled the home crowd, becoming more and more threatening when Andros Townsend slipped in Kyle Walker down the right and his precise centre was met by Roberto Soldado. Soldado's hit on goal was bravely charged down by Per Mertesacker.
A period of relative calm was followed by a home goal on 23 minutes. On the break Rosicky scurried forward and played in Walcott in space down the right flank. With a lot of time to pick out his pass the England player poked towards the near post where Giroud time his run to prod past an exposed Lloris.
Having conceded a first goal of the season, Lloris was then forced to stoop low to thwart a Walcott drilled effort after he again made capital down the right flank.
Townsend then forced Wojciech Szczesny into a smart save down to his left after the wide man cut inside from the right and fired in with considerable power.
Lloris turned centre half outside his area, executing a high-risk, yet vital, challenge on a flying Walcott. The goalkeeper got the ball despite Walcott's tumble but had he not Brad Friedel could have been doing a speedy warm-up.
Arsenal appeared to have the momentum as the clock ticked towards half-time, certainly carrying a greater threat on quick counter-attacks, with the pace of Walcott and the promptings of Rosicky and Cazorla. The team needed to find a way of planting the home side on the back foot and keeping them there. A dash of dynamism was required as Arsenal were quickly getting men behind the ball when not in possession.
Townsend kept trying and again worked Szczesny with a drive from outside the area.
Arsene Wenger was forced into a change before the break when Jack Wilshere appeared to be struggling, which meant a sooner than expected return for Mathieu Flamini to this occasion.
Walker's corner caused a degree of panic in the opening minutes of the second half as the home rearguard failed to deal effectively with his low set piece. Nacer Chadli flighted over a free-kick after Danny Rose was upended by Flamini, but Andre would have been hoping for more in-roads to have been made into home territory in the first third of the second half.
A devilish cross from Townsend so nearly fell kindly for Chadli at the far post but a defensive glance quashed what was a promising move as the team moved up through the gears and started to show some verve.
The good work was nearly undone when Giroud broke through at the other end and only a last-ditch challenge by Rose knocked him out of his stride and resulted in him shooting wide, which was what Paulinho did moments later from 20-odd yards.
It was enter Jermain Defoe on 69 minutes in place of Dembele, but it was Lloris next in action after Michael Dawson lost possession and the Frenchman sprinted off his line to brilliantly deny Walcott.
Capoue sustained what looked like an ankle injury in a tangle with Cazorla, which led to the introduction of Sandro along with Lamela for his debut with 16 minutes left on the clock - Townsend being the other to make way.
There was a mighty scramble straight away when a deflected Defoe shot was saved excellently by Szczesny before Soldado claimed a hand ball when he attempted to smash home the rebound after Paulinho picked up the loose ball.
Lloris foiled Walcott in what was a big Arsenal chance with 10 minutes left, with sub Nacho Monreal failing to connect with the rebound, and the hope was the one that got away would be costly.
Defoe picked out Lamela with six minutes left and the new boy turned well before shooting tamely at Szczesny.
The team pressed on until the end but despite a couple of scrambles in the Arsenal box, it was not to be.
Arsenal (4-3-3): Szczesny; Jenkinson, Mertesacker, Koscielny, Gibbs; Ramsey, Wilshere (Flamini, 43), Rosicky (Monreal, 79); Cazorla, Giroud, Walcott (Sagna, 90+2). Subs not used: Fabianski, Sanogo, Gnabry, Zelalem.
Spurs (4-2-3-1): Lloris; Walker, Dawson, Vertonghen, Rose; Capoue (Sandro, 75), Paulinho; Townsend (Lamela, 75), Dembele (Defoe, 69), Chadli; Soldado. Subs not used: Friedel, Naughton, Holtby, Sigurdsson.
Goal: Arsenal - Giroud (23).
Yellow cards: Arsenal - Rosicky, Flamini. Spurs - Defoe.
Referee: Mr M Oliver.