  Let $P(K)$ be a satellite knot where the pattern, $P$, is a Berge-Gabai knot
(i.e., a knot in the solid torus with a non-trivial solid torus Dehn surgery),
and the companion, $K$, is a non-trivial knot in $S^3$. We prove that $P(K)$ is
an L-space knot if and only if $K$ is an L-space knot and $P$ is sufficiently
positively twisted relative to the genus of $K$. This generalizes the result
for cables due to Hedden and the first author.
