Stilbenes are plant defense compounds known to rapidly accumulate in grapevine and some other plant species in response to microbial infection and several abiotic stresses. Stilbenes have attracted considerable attention due to valuable biological effects with multi-spectrum therapeutic application. However, there is a lack of information on natural signaling pathways and transcription factors regulating stilbene biosynthesis. It has been previously shown that MYB R2R3 transcription factor genes VaMyb40 and VaMyb60 were up-regulated in cell cultures of wild-growing grapevine Vitis amurensis Rupr. in response to UV irradiation. In this study, the effects of VaMyb40 or VaMyb60 overexpression in cell cultures of V. amurensis on their capability to produce stilbenes were investigated. Overexpression of the VaMyb60 gene led to a considerable increase in the content of stilbenes in three independently transformed transgenic lines in 5.9–13.9 times, while overexpression of the VaMyb40 gene also increased the content of stilbenes, although to a lesser extent (in 3.4–4.0 times) in comparison with stilbene levels in the empty vector-transformed calli. Stilbene content and stilbene production in the VaMyb60-transgenic calli reached 18.8 mg/g of dry weight (DW) and 150.8 mg/L, respectively. Using HPLC analysis, we detected eight individual stilbenes: t-resveratrol diglucoside, t-piceid, t-resveratrol, ε-viniferin, δ-viniferin, cis-resveratrol, cis-piceid, t-piceatannol. T-resveratrol prevailed over other stilbenoid compounds (53.1–89.5% of all stilbenes) in the VaMyb-overexpressing cell cultures. Moreover, the VaMyb40- and VaMyb60-transformed calli were capable of producing anthocyanins up to 0.035 mg/g DW, while the control calli did not produce anthocyanins. These findings show that the VaMyb40 and VaMyb60 genes positively regulate the stilbene biosynthesis as strong positive transcription regulators and can be used in biotechnological applications for stilbene production or high-quality viticulture and winemaking.