© PA Photos
Giroud opened the scoring on 16 minutes, confidently dispatching a Lukas Podolski layoff into the top corner moments after Ashley Barnes had tested Wojciech Szczesny with the first effort of the match.
Barnes was not to be denied later in the half as he met a David Lopez corner with a close-range header, which left the stranded Szczesny with no chance.
Giroud recorded his fourth double of the season when he collected a precise chipped through-ball from Abou Diaby and swept home after the interval.
Six minutes later, though, debutante striker Leonardo Ulloa, who had been denied a first-half goal by the offside flag, opened his Seagulls account by bravely stooping to head Inigo Calderon's cross into the net.
The Premier League visitors pushed for a third goal and came close when Podolski rattled the crossbar with a powerful free kick.
The woodwork was not on hand to save Brighton when Walcott slotted the ball into the bottom corner with five minutes of time remaining.