Axelrod’s model for the dissemination of culture contains two key factors required to model the process of diffusion of innovations, namely, social influence (i.e., individuals become more similar when they interact) and homophily (i.e., individuals interact preferentially with similar others). The strength of these social influences are controlled by two parameters: F, the number of features that characterizes the cultures and q, the common number of states each feature can assume. Here we assume that the innovation is a new state of a cultural feature of a single individual – the innovator – and study how the innovation spreads through the networks among the individuals. For infinite regular lattices in one and two dimensions, we find that initially the innovation spreads linearly with the time t and diffusively in the long time limit, provided its introduction in the community is successful. For finite lattices, the growth curves for the number of adopters are typically concave functions of t. For random graphs with a finite number of nodes N, we argue that the classical S-shaped growth curves result from a trade-off between the average connectivity K of the graph and the per feature diversity q. A large q is needed to reduce the pace of the initial spreading of the innovation and thus delimit the early-adopters stage, whereas a large K is necessary to ensure the onset of the take-off stage at which the number of adopters grows superlinearly with t. In an infinite random graph we find that the number of adopters of a successful innovation scales with tγ with γ=1 for K>2 and ½<γ<1 for K=2. We suggest that the exponent γ may be a useful index to characterize the process of diffusion of successful innovations in diverse scenarios.