The Football Association upheld the decision of referee Andre Marriner to dismiss the France international in the final minute of Saturday's 1-0 win at Craven Cottage for jumping into a challenge with defender Stanislav Manolev.
As a result, Giroud must now serve the mandatory suspension - which starts against Manchester United on Sunday.
A statement from the FA read: "Following a Regulatory Commission hearing, Arsenal's Olivier Giroud has had his claim for wrongful dismissal rejected.
"Giroud was dismissed for serious foul play during his side's game against Fulham at Craven Cottage on Saturday 20 April 2013. The player's three-match suspension will therefore commence with immediate effect."
The loss of Giroud, who has netted 17 goals since his £13million summer move from Montpellier, is certainly a blow to Wenger's squad as they look to lock-down Champions League qualification. Arsenal retain a slender one-point lead over fourth-placed Chelsea and are two ahead of north London rivals Tottenham, albeit having played a match more.
As well as the showdown with United - when the Gunners are expected to form a guard of honour for the newly-crowned champions - Giroud must also sit out the trip to relegation-threatened QPR and the final home match of the season against FA Cup finalists Wigan before he will be eligible again on the last day at Newcastle.
Theo Walcott is one option for a central striker, and is currently leading scorer with 18, while German international Lukas Podolski is another who has been pressing for a start down the middle, looking to add to a tally of 14.
With Chelsea and Tottenham set to face each other at Stamford Bridge on May 8, Arsenal know if they maintain the momentum of five wins from their previous six league games, then a top-four finish will be within reach.
Arsenal have responded magnificently since they were beaten 2-1 at Tottenham on March 3, with their fine run of form kicked off by winning at Bayern Munich, albeit losing the Champions League tie on away goals.