This works best via the theorem on partitions (Theorem 7.2.15 in Engelking's General Topology): $\dim X\le n$ iff for every sequence $(A_1,B_1)$, ..., $(A_n,B_n)$, $(A_{n+1},B_{n+1})$ of pairs of disjoint closed sets there is a sequence $(U_1,V_1)$, ..., $(U_n,V_n)$, $(U_{n+1},V_{n+1})$ of pairs of disjoint open sets, with $A_i\subseteq U_i$ and $B_i\subseteq V_i$ for all $i$, such that $\bigcap_{i=1}^{n+1}L_i=\emptyset$, where $L_i=X\setminus(U_i\cup V_i)$.

Your condition implies $\dim X\le n$ by considering continuous $f_i:X\to[-1,1]$ with $f_i[A_i]=\{-1\}$ and $f_i[B_i]=\{1\}$; use a function $g$ that is closer than $1/3$ to $(f_1,\ldots,f_n,f_{n+1})$ and that avoids the origin to create the $U_i$ and $V_i$.

Conversely, given $f$ let $\varepsilon>0$ and consider the closed sets $A_i=\{x:f_i(x)\le-\varepsilon\}$ and $B_i=\{x:f_i\ge\varepsilon\}$; find the $U_i$, $V_i$ and $L_i$ and use Urysohn functions $g_i$ that satisfy $g[A_i]=\{-\varepsilon\}$, $g_i[L_i]=\{0\}$ and $g_i[B_i]=\{\varepsilon\}$ to change $f$ outside the union of the $A_i$s and $B_i$s so as to avoid the origin.

For a concrete case look at maps from the unit interval to the plane; these can be approximated uniformly by piecewise linear maps and it is always possible to change the latter so as to avoid the origin. This can even be done with a surjective map from $[-1,1]$ onto the square $[-1,1]^2$.