N6-methyladenosine (m6A) is the most abundant eukaryotic modification internal mRNA, which plays the crucial roles in the occurrence and development of cancer. However, current knowledge about m6A-mediated functional circuit and key genes targeted by m6A methylation in cancer is mostly elusive. Thus, here we proposed a novel network-based approach (called m6Acancer-Net) to identify m6A-mediated driver genes and their associated network in specific type of cancer, such as acute myeloid leukemia. m6A-mediated cancer driver genes are defined as genes mediated by m6A methylation, significantly mutated, and functionally interacted in cancer. m6Acancer-Net identified the m6A-mediated cancer driver genes by combining gene functional interaction network with RNA methylation, gene expression and mutation information. A cancer-specific gene-site heterogeneous network was firstly constructed by connecting the m6A site co-methylation network with the functional interaction pruned gene co-expression network generated from large scale gene expression profile of specific cancer. Then, the functional m6A-mediated genes were identified by selecting the m6A regulators as seed genes to perform the random walk with restart algorithm on the gene-site heterogeneous network. Finally, m6A-mediated cancer driver gene subnetworks were constructed by performing the heat diffusion of mutation frequency for functional m6A-mediated genes in protein-protein interaction networks. The experimental results of m6Acancer-Net on the acute myeloid leukemia (AML) and glioblastoma multiforme (GBM) data from TCGA project show that the m6A-mediated caner driver genes identified by m6Acancer-Net are targeted by m6A regulators, and mediate significant cancer-related pathways. They play crucial roles in development and prognostic stratification of cancer. Moreover, 15 m6A-mediated cancer driver genes identified in AML are validated by literatures to mediate AML progress, and 14 m6A-mediated cancer driver genes identified in GBM are validated by literatures to participate in development of GBM. m6Acancer-Net is reliable to identify the functionally significant m6A-mediated driver genes in specific cancer, and it can effectively facilitate the understanding of regulatory and therapeutic mechanism of cancer driver genes in epitranscriptome layer.