  We consider a class of sparse random matrices which includes the adjacency
matrix of the Erd\H{o}s-R\'enyi graph $\mathcal{G}(N,p)$. We show that if
$N^{\varepsilon} \leq Np \leq N^{1/3-\varepsilon}$ then all nontrivial
eigenvalues away from 0 have asymptotically Gaussian fluctuations. These
fluctuations are governed by a single random variable, which has the
interpretation of the total degree of the graph. This extends the result [19]
on the fluctuations of the extreme eigenvalues from $Np \geq N^{2/9 +
\varepsilon}$ down to the optimal scale $Np \geq N^{\varepsilon}$. The main
technical achievement of our proof is a rigidity bound of accuracy
$N^{-1/2-\varepsilon} \, (Np)^{-1/2}$ for the extreme eigenvalues, which avoids
the $(Np)^{-1}$-expansions from [9,19,24]. Our result is the last missing
piece, added to [8, 12, 19, 24], of a complete description of the eigenvalue
fluctuations of sparse random matrices for $Np \geq N^{\varepsilon}$.
