With Bale's world record move to Real Madrid set to be concluded before Monday evening's transfer deadline, Spurs have embarked on a frenzied spending spree.
Villas-Boas has splashed out over £100 million on seven players and four of those new arrivals, Roberto Soldado, Paulinho, Etienne Capoue and Nacer Chadli, were in the starting line-up at the Emirates.
But, with little time to gel, Tottenham lacked cohesion and unity in their first serious test of the post-Bale era.
Spurs were under duress from the start and Santi Cazorla's free-kick brought a fine save from Hugh Lloris with just two minutes gone.
Arsenal were playing with an intuitive understanding their rivals badly missed and they took the lead in the 23rd minute with the kind of immaculately constructed goal that makes Wenger's teams such a joy to watch when they hit top gear.
An Aaron Ramsey pass located Tomas Rosicky in space on the right of midfield and the Czech veteran quickly shifted possession onto Walcott, who surged into the penalty area and whipped over a low cross that French striker Giroud, fatally granted too much space by Michael Dawson, met with a superb flicked finish past Lloris from close-range.
There was no let-up from Arsenal and Ramsey showed tremendous determination to fight his way past Mousa Dembele before slipping a pass into Walcott, whose shot was pushed away by Lloris.
Lloris was then forced to race out of his area to make a last-ditch tackle on Walcott as the England winger surged clear of Tottenham's alarmingly flat-footed defence.
Walcott crashed to the turf, drawing inevitable Arsenal appeals for Lloris's dismissal, but referee Michael Oliver waved play on and replays showed he got the decision spot on.
Deprived of Bale's blistering pace and lethal finishing, this was a Tottenham side sorely lacking inspiration.
Chadli, Paulinho and Capoue were lacklustre in midfield, while Soldado was starved of service up front.
The loss of England midfielder Jack Wilshere, forced off just before half-time due to an illness, gave Mathieu Flamini the chance for his second debut after returning to Arsenal on a free transfer and offered Spurs hope of a second half fightback.
But it was Arsenal who remained the more potent force and Giroud went close when his shot deflected wide off Danny Rose.
Villas-Boas sent on Argentine forward Eric Lamela for his debut following his move from Roma in the closing stages, while Capoue was stretchered off wearing an oxygen mask after falling awkwardly.
Tottenham finally threatened an equaliser when Wojciech Szczesny pushed Jermain Defoe's deflected strike out to Soldado and the Spaniard lashed in a shot that drew rejected penalty claims for a handball by Carl Jenkinson.
That was the signal for Spurs to throw bodies forward in the frantic final minutes, but Arsenal held firm to inflict their neighbours' first defeat of the season.