We fabricated organic field-effect transistors (OFETs) having a thin layer of molybdenum trioxide (MoO$_3$), a Lewis acid, and evaluated their electrical characteristics. The insertion of a thin MoO$_3$ layer reduces the on/off ratio but improves the apparent mobility of the charge carriers. To identify the dominant mechanism responsible for this effect, we characterized devices having a 69-nm-thick pentacene layer with a 1-nm-thick MoO$_3$ layer either between the gold source and the drain electrodes or only directly under these electrodes. The former device exhibited a low on/off ratio, whereas the latter device exhibited an on/off ratio comparable to those of conventional pentacene OFETs without a thin MoO$_3$ layer, suggesting that the formation of charge-transfer (CT) complexes immediately above the conduction channel is the critical mechanism. CT complexes at the pentacene/MoO$_3$ interface immediately above the conduction channel contribute to the formation of an effective channel for off-currents as well as drain currents. Moreover, we also attempted to improve the on/off ratio by using a cloth to rub the surface of a thin MoO$_3$ layer immediately above the conduction channel to create what we believe to be a profile with abrupt changes in height in the direction of the drain current conduction in OFETs. Consequently, it was found that such a rubbed MoO$_3$ layer had a surface with a scratched pattern, and the on/off ratio of the OFET was improved, indicating that controlling the CT complex formation by patterning a MoO$_3$ layer can reduce the off-current in OFETs having a pentacene/MoO$_3$ active layer.