We prove the existence of a weak solution for a general class of Dirichlet anisotropic elliptic problems such as $\mathcal{A}u+\mathrm{\Phi }(x,u,\mathrm{\nabla }u)=\mathfrak{B}u+f$ in $\mathrm{\Omega }$, where $\mathrm{\Omega }$ is a bounded open subset of ${\mathbb{R}}^N$ and $f\in L^1(\mathrm{\Omega })$ is arbitrary. The principal part is a divergence-form nonlinear anisotropic operator $\mathcal{A}$, the prototype of which is $\mathcal{A}u=-\sum _{j=1}^N{\mathrm{\partial }}_j(|{\mathrm{\partial }}_{j}u{|}^{p_j-2}{\mathrm{\partial }}_{j}u)$ with $p_j>1$ for all $1\le j\le N$ and $\sum _{j=1}^N(1/p_j)>1$. As a novelty in this paper, our lower order terms involve a new class of operators $\mathfrak{B}$ such that $\mathcal{A}-\mathfrak{B}$ is bounded, coercive and pseudo-monotone from $W_0^{1,\overrightarrow{p}}(\mathrm{\Omega })$ into its dual, as well as a gradient-dependent nonlinearity $\mathrm{\Phi }$ with an "anisotropic natural growth" in the gradient and a good sign condition.