The answer is yes -- Morton Brown's mapping theorem says that for every closed (connected) topological $n$-manifold $M$ there is a continuous map $f$ from the $n$-cube $I^n$ onto $M$ which is injective on the interior of the cube (for manifolds with boundary, see Remark 1 below). This was proved in early sixties and can be found in M.Brown,*"A mapping theorem for untriangulated manifolds,"* Topology of 3-manifolds, pp.92-94, M.K.Fort, Jr.(Editor), Prentice-Hall, Englewood Cliffs, N.J.,1962. MR0158374

Main idea of the proof is simple: "Use local PL structures to expand a small $n$-cell in $M$ gradually, until it becomes the whole manifold."

This can be realized the "infinite composition" of engulfings of finitely many points at a time. This argument is not too difficult, so I will try to sketch it.

First consider a closed $n$-cell $C$ in $M$ and a finite set $X=\{x_1,\ldots,x_k\}$ of points of $M$ disjoint from $C$ which we want to engulf. Assume that each $x_i$ lies in some open $n$-cell $U_i$ which intersects $C$. For each $i$, fix a PL structure on $U_i$, and join $x_i$ and some point $y_i\in \partial C$ by a PL arc $\alpha_i\subset U_i$ relative to this structure. If we assume $\dim M\geq 3$ (this is not a restriction because the theorem is clear for $\dim M=1$, and easily follows from the surface classification for $\dim M=2$), we may require that $\alpha_i$'s should be disjoint. The regular neighborhood $Q_i$ of $\alpha_i$ within $U_i$ is a closed $n$-cell, and $Q_i$'s can be made disjoint. Let $h$ be a homeomorphism of $M$ pushing $y_i$ towards $x_i$ within $Q_i$ for each $i$ which is identity outside $Q_i$'s. Then $X\subset h(C)$, and we can apply this process again to $h(C)$ and another finite set $X'\subset M\setminus h(C)$. Repeating this process we obtain a sequence of engulfing homeomorphisms $h_1, h_2,\ldots$.

We can arrage that the uniform limit $f\colon M\to M$ of the composition of engulfing homeomorphisms $f_n=h_n\circ\cdots\circ h_1$ exists and $f(C)=M$. If we choose sufficiently small $Q_i$'s in each stage, one can make sure that $f$ is injective on the interior of $C$ (indeed, we can arrange that for each interior point $x$ there is $n$ such that $f_n(x)=f(x)$).

Remark 1: In Brown's paper, one can find a corollary that "If $M$ is a compact
connected manifold with nonempty boundary $B$, then there is a surjection $f\colon B\times [0,1]\to M$ that restricts to the identity on $B\times 0$ and is injective on $B\times[0,1)$". Notice also that the above theorem can be applied to the **closed** manifold $B$. Then it follows that any compact **topological** $n$-manifold (possibly with boundary) can be obtained by identifying some points in the boundary of the $n$-ball.

Remark 2: Berlanga's theorem extended the Brown's theorem to noncompact manifolds:
This theorem states that for every (connected) $\sigma$-compact $n$-manifold $M$, there is a nice kind of surjection $I^n\to \overline{M}$ similar to Brown's map, where $\overline{M}$ denotes the end compactification of $M$.