Over the past decade, the richness of electronic properties of graphene has attracted enormous interest for electrically detecting chemical and biological species using this two-dimensional material. However, the creation of practical graphene electronic sensors greatly depends on our ability to understand and maintain a low level of electronic noise, the fundamental reason limiting the sensor resolution. Conventionally, to reach the largest sensing response, graphene transistors are operated at the point of maximum transconductance, where 1/f noise is found to be unfavorably high and poses a major limitation in any attempt to further improve the device sensitivity. We show that operating a graphene transistor in an ambipolar mode near its neutrality point can markedly reduce the 1/f noise in graphene. Remarkably, our data reveal that this reduction in the electronic noise is achieved with uncompromised sensing response of the graphene chips and thus significantly improving the signal-to-noise ratio-compared to that of a conventionally operated graphene transistor for conductance measurement. As a proof-of-concept demonstration of the usage of the aforementioned new sensing scheme to a broader range of biochemical sensing applications, we selected an HIV-related DNA hybridization as the test bed and achieved detections at picomolar concentrations.