This is not necessarily true.

Consider $M = \mathbb{R}^2$ less three points (say $-1, 0, 1$), and let $Z_1$ be a loop around $-1$ and $0$, while $Z_2$ is a loop around 0 and 1. Then their index of intersection is 0, but they cannot be replaced by cycles which do not intersect.

It is of course worth noting that $M$ is not compact in this case.

Edit: Nevermind, this is not true at all if $Z_i$ do not have to be connected, which you never said they must be.

Edited:

If you require the $Z_i$ to be connected, then this is not true.

Consider $M = \mathbb{R}^2$ less three points (say $-1, 0, 1$), and let $Z_1$ be a loop around $-1$ and $0$, while $Z_2$ is a loop around 0 and 1. Then their index of intersection is 0, but they cannot be replaced by cycles which do not intersect.

It is of course worth noting that $M$ is not compact in this case.

However, if the $Z_i$ are allowed to be disconnected, then the argument I just gave falls apart. Either of the $Z_i$ are homologous to a disjoint union of two circles around their respective centres, and so it is clear that they can be made to be disjoint.