The stochastic block model (SBM), a popular framework for studying community detection in networks, is limited by the assumption that all nodes in the same community are statistically equivalent and have equal expected degrees. The degree-corrected stochastic block model (DCSBM) is a natural extension of SBM that allows for degree heterogeneity within communities. To find the communities under DCSBM, this paper proposes a convexified modularity maximization approach, which is based on a convex programming relaxation of the classical (generalized) modularity maximization formulation, followed by a novel doubly-weighted $\ell_{1}$-norm $k$-medoids procedure. We establish nonasymptotic theoretical guarantees for approximate and perfect clustering, both of which build on a new degree-corrected density gap condition. Our approximate clustering results are insensitive to the minimum degree, and hold even in sparse regime with bounded average degrees. In the special case of SBM, our theoretical guarantees match the best-known results of computationally feasible algorithms. Numerically, we provide an efficient implementation of our algorithm, which is applied to both synthetic and real-world networks. Experiment results show that our method enjoys competitive performance compared to the state of the art in the literature.