Giroud's header in the 86th minute provided a welcome return to winning ways for Arsenal after painful defeats on the road.
Arsene Wenger's side seemed certain to slip further off the pace at the top of the league against an ultra-defensive Albion team as the game moved into its final stages at the Emirates Stadium.
But Giroud's header in the 86th minute provided a welcome return to winning ways for Arsenal after painful defeats on the road at Everton and Manchester City.
Gunners boss Wenger had defended Mesut Ozil, a scapegoat in some quarters for failing to pull his defensive weight in those pre-Christmas setbacks.
And he was rewarded for his faith as Ozil provided the cross for Giroud's winner to keep Arsenal nine points behind leaders Chelsea.
It was clear from early on that Albion's priority was keeping a tight rein on Arsenal's attack and most of the first half followed a repeated pattern -- the hosts moving the ball forward and attempting to find gaps without much success.
Ozil had a shot blocked, Giroud sent a curling effort high and wide, and then failed to react quickly enough to provide a target when Hector Bellerin beat the offside trap and volleyed the ball across the penalty area.
When Arsenal attempted to pass their way through, any inaccuracy or failure of control was punished as defenders surrounded and outnumbered attackers.
Claudio Yacob intercepted as Alex Iwobi and Giroud attempted an intricate one-two.
Iwobi then decided to go it alone, but his shot drifted just over the bar after he had dribbled in from the left.
In the opening five minutes of the second half Arsenal raised the pace and urgency of their game and might have scored twice.
First Iwobi's low volley forced a save from Albion goalkeeper Ben Foster and as Alexis Sanchez rushed in to attempt to stab home the loose ball, Foster knocked it away for a corner.
From the resulting set piece, the ball reached Sanchez inside the six-yard box but with his back to goal.
The Chilean dribbled away from goal to his left and shot on the turn but the ball rebounded off the near post.
Sanchez, full of attacking intent, then ran in from the left and shot fiercely for the top corner, his drive finger-tipped away by Foster.
Perhaps that inspired Giroud to run past two defenders and hit a low, left-foot shot that Foster had to keep out with his leg.
Albion had little to offer until Salomon Rondon charged down the left, but Gabriel was able to outpace him as he bore down on goal.
A second run by Rondon, however, forced the first of a series of corners that ended with Yacob blasting over the crossbar after Petr Cech had failed to make a clean catch.
At the other end Giroud's outstretched foot was inches away from reaching a low cross from Lucas Perez, and when Bellerin worked his way clear of defenders to meet a cross from the right, he could only head the ball over the bar.
It seemed Albion had done enough to earn a point until Ozil crossed from the right and Giroud, bumping and barging defender Gareth McAuley, sent a header looping over Foster and in.
Late substitute Aaron Ramsey almost made it 2-0 when he ran onto Sanchez's header only to hit his shot against Foster's legs.