The Frenchman met Santi Cazorla's left-wing corner with a powerful downward header in the 14th minute, ensuring Arsene Wenger's men picked up their fourth league win in a row.
Everton's 3-2 reverse to Manchester City on Saturday meant Arsenal were already assured of a UEFA Champions League berth for a 17th successive campaign, with the FA Cup final against Hull City on May 17 now their main focus.
West Brom, meanwhile, went into the game knowing they had likely done enough to secure safety, with 18th-placed Norwich City facing two difficult fixtures, and there was little intensity on show in a low-key affair.
Lukas Podolski did strike the post for Arsenal in the second half, but one goal proved enough as West Brom struggled to carve out opportunities.
Arsenal hit the front just before the quarter-hour mark thanks to Giroud's firm header.
After Cazorla had seen his shot deflected over the bar by Youssouf Mulumbu, the Spaniard curled the resulting corner in from the left and Giroud got the better of Craig Dawson to plant a well-directed header into the net from six yards.
Graham Dorrans stung the palms of Wojciech Szczesny with a rasping drive in the 32nd minute, while Saido Berahino blasted a shot narrowly over a minute later.
The hosts' response was immediate, Cazorla bringing a smart save out of Ben Foster after running onto Podolski's cut-back.
Mathieu Flamini, back in the side in place of the rested Aaron Ramsey, hit a tame effort straight at Foster from the edge of the box as Arsenal ended the first period in the ascendancy.
In a quiet second half, Dawson headed over the bar from a corner in the 72nd minute before dragging a speculative effort well wide of goal from 30 yards shortly afterwards.
Mesut Ozil, quiet for much of the match, stabbed narrowly wide of goal after getting on the end of Giroud's flick as Arsenal looked to put the game to bed.
Podolski's mis-hit shot then rebounded back off the post and into the arms of a grateful Foster, but it mattered little.
While Arsenal's title hopes faded some time ago, the club can at least take some consolation from reaching their highest Premier League points tally since the 2007-08 season.
