For a class of sparse random matrices of the form $A_n =(ξ_{i,j}δ_{i,j})_{i,j=1}^n$, where $\{ξ_{i,j}\}$ are i.i.d.~centered sub-Gaussian random variables of unit variance, and $\{δ_{i,j}\}$ are i.i.d.~Bernoulli random variables taking value $1$ with probability $p_n$, we prove that the empirical spectral distribution of $A_n/\sqrt{np_n}$ converges weakly to the circular law, in probability, for all $p_n$ such that $p_n=ω({\log^2n}/{n})$. Additionally if $p_n$ satisfies the inequality $np_n > \exp(c\sqrt{\log n})$ for some constant $c$, then the above convergence is shown to hold almost surely. The key to this is a new bound on the smallest singular value of complex shifts of real valued sparse random matrices. The circular law limit also extends to the adjacency matrix of a directed Erdős-Rényi graph with edge connectivity probability $p_n$.