Both keepers had to be sharp in an open North London derby but especially Tottenham's Hugo Lloris, who denied Walcott and Giroud.
Tottenham poured forward in a frantic finish, though couldn't find an equalizer.
Spurs and Arsenal now have identical records in the league at two wins and a loss.
While Tottenham has made seven signings this summer in anticipation of Bale's probable transfer to Real Madrid, Arsenal has merely brought in two free transfers even with ample funds available.
Manager Arsene Wenger, however, hinted it could be a busy Monday.
"Maybe we'll have a good surprise for you," he told Sky Sports. "We're working very hard."
In the lone other game in the Premier League, Swansea recorded its first league win by downing still-winless West Bromwich Albion 2-0 away.
French striker Nicolas Anelka returned for West Brom after taking time off due to the death of his agent.