This paper establishes Carleson embeddings of Müntz spaces $M^q_{\Lambda}$ into weighted Lebesgue spaces $L^p(\mathrm{d}\mu)$, where $\mu$ is a Borel regular measure on $[0,1]$ satisfying $\mu([1-\varepsilon])\lesssim \varepsilon^{\beta}$. In the case $\beta \geqslant 1$ we show that such measures are exactly the ones for which Carleson embeddings $L^{\frac{p}{\beta}} \hookrightarrow L^p(\mathrm{d}\mu)$ hold. The case $\beta \in (0,1)$ is more intricate but we characterize such measures $\mu$ in terms of a summability condition on their moments. Our proof relies on a generalization of $L^p$ estimates à la Gurariy-Macaev in the weighted $L^p$ spaces setting, which we think can be of interest in other contexts.