Let $\mathfrak{ g}$ be a curved $L_\infty$-algebra endowed with a complete filtration $\mathfrak{ F}\mathfrak{ g}$. Suppose there exists an integer $r \in \mathbb{ N}_0$ for which the curvature $\mu_0$ satisfies $\mu_0 \in \mathfrak{ F}_{ 2r+1} \mathfrak{ g}$ and the spectral sequence yields $E_{ r+1}^{ p,q} =0$ for $p,q$ with $p+q=2$. We prove that then a Maurer-Cartan element exists. In addition, we show, as a typical application, that for $P$ a possibly inhomogeneous Koszul operad with generating set in arities 1,2 (e.g. $P$=Com,As,BV,Lie,Ger), a $P_\infty$-algebra $A$ is intrinsically formal if its twisted deformation complex $\mathrm{ Def}(H(A)\stackrel{ \mathrm{ id}}{ \to} H(A))$ is acyclic in total degree 1.
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