Not sure if this is what you are looking for, but a homeomorphism that maps zero sets to zero sets in both directions will preserve Lebesgue Measurability.
Alternately, a homeomorphism which is Lipschitz in both directions (a so-called Lipeomorphism) preserves Lebesgue Measurability. This is because Lipschitz maps take zero sets to zero sets.
Finally, diffeomorphisms in general preserve Lebesgue Measurability. This is because they are locally Lipschitz. This follows from the $n-$dimensional Mean Value Theorem.
Also, it is not true that the continuous image of a zero set is a zero set. The Cantor Set is a zero set in $\mathbb{R}$, but yet there exists a continuous surjection of the Cantor Set onto any compact metric space.