The upper tail problem in the Erdős--Rényi random graph $G\sim\mathcal{G}_{n,p}$ asks to estimate the probability that the number of copies of a graph $H$ in $G$ exceeds its expectation by a factor $1+δ$. Chatterjee and Dembo showed that in the sparse regime of $p\to 0$ as $n\to\infty$ with $p \geq n^{-α}$ for an explicit $α=α_H>0$, this problem reduces to a natural variational problem on weighted graphs, which was thereafter asymptotically solved by two of the authors in the case where $H$ is a clique. Here we extend the latter work to any fixed graph $H$ and determine a function $c_H(δ)$ such that, for $p$ as above and any fixed $δ>0$, the upper tail probability is $\exp[-(c_H(δ)+o(1))n^2 p^Δ\log(1/p)]$, where $Δ$ is the maximum degree of $H$. As it turns out, the leading order constant in the large deviation rate function, $c_H(δ)$, is governed by the independence polynomial of $H$, defined as $P_H(x)=\sum i_H(k) x^k $ where $i_H(k)$ is the number of independent sets of size $k$ in $H$. For instance, if $H$ is a regular graph on $m$ vertices, then $c_H(δ)$ is the minimum between $\frac12 δ^{2/m}$ and the unique positive solution of $P_H(x) = 1+δ$.