Arsenal's troubles deepened in the second half when Dimitar Berbatov put Fulham ahead from the penalty spot.
Giroud equalised for Arsenal with his second headed goal and the hosts could have won the match in stoppage time but Mikel Arteta's penalty was saved.
Elsewhere, Adam Johnson ended Sunderland's eight-hour and 55-minute goal drought but goals from Marouane Fellaini and Nikica Jelavic gave Everton a 2-1 win at Goodison Park on Saturday. England winger Johnson, 25, prodded in just before the interval to score his first goal since joining the club from Manchester City in a £10 million ($15.9 million, 12.5 million euros) deal in August.
But just as Sunderland looked set to cling on for only a second win in 18 matches, Fellaini pulled Everton level with his sixth Premier League goal of the season with 15 minutes left.
Less than three minutes later, Jelavic added a second with his fifth of the campaign to make sure the Toffees extended their unbeaten run against the Black Cats to 16 matches.