Graphene has attracted much attention for sensing applications in recent years. Its largest surface-to-volume ratio makes graphene sensors able to potentially detect a single molecule and its extremely high carrier mobility ensures low electrical noise and energy consumption. However, pristine graphene is chemically inert and weakly adsorbs gas molecules, while defective and/or doped graphene has stronger adsorption ability (high sensitivity). The high sensitivity is related to the increased number of defects or traps in graphene where the gas molecules can be readily grafted, changing the sensor resistance. Nonetheless, similar resistance changes could be induced under exposure to different gases, resulting in a lack of selectivity. Functional groups differ drastically from defects or traps since the former selectively anchor specific molecules. Here, we comparatively investigate three functionalization routes and optimize a defect-free one (2,3,5,6,-Tetrafluorohydroquinone, TFQ molecules) for the fabrication of graphene gas sensors. We use TFQ organic molecules as chemical recognition links between graphene and formaldehyde, the most common indoor pollutant gas. The sensor demonstrates a high response and a good selectivity for formaldehyde compared with interfering organic vapours. Particularly, the sensor has a strong immunity to humidity. Our results highlight that defect-free functionalization based on organic molecules not only increases the sensor's response but also its selectivity, paving the way to the design of efficient graphene-based sensors.