Surjective endomorphisms of finitely generated modules are isomorphisms
Yes, a surjective $A$-linear endomorphism $T:M\to M$ of a finitely generated $A$-module $M$ is an isomorphism.
Proof
Consider $M$ as an $A[X]$-module via the multiplication $P(X)*m=P(T)(m)$ , so that for example $(X^2-7)*m=T(T(m))-7m$.
[this is a classical trick used in advanced linear algebra].
Surjectivity of $T$ translates into $M=XM$ and so a fortiori for the ideal $I=XA[X]$ we have $M=IM$.
Now Nakayama's lemma comes to our rescue : it says that there exists an element $XQ(X)\in I$ such that $(1-XQ(X))*m=0$ for all $m\in M$, which means that $m=TQ(T)m$ and this immediately implies that $T$ is invertible with inverse $T^{-1}=Q(T)$.
This result and its extremely elegant proof are due to Vasconcelos.
And if, like so many of us , you keeep forgetting what Nakayama says, look here.
Caveat
Of course an injective endomorphism of a finitely generated module needn't be surjective: take $A=\mathbb Z$ and $T:\mathbb Z\to\mathbb Z:m\mapsto 2m$ !
The following proof is based on the paper: "Onto Endomorphisms are Isomorphisms", Amer. Math. Monthly 78 (1971), 357-362 by Morris Orzech, Queen's University.
I was told about this paper by KCd in this thread.
The idea is to reduce the theorem to an easy case where $A$ is a Noetherian ring.
Lemma (a slight generalization of Atiyah-Macdonald's Exercise 6.1) Let $A$ be a not-necessarily commutative ring. Let $M$ be a Noetherian $A$-module. Let $N$ be an $A$-submodule of $M$. Let $f\colon N \rightarrow M$ be a surjective $A$-homomorphism. Then $f$ is injective.
Proof Let $K_n = \ker(f^n)$, $n = 1, 2,\dots$. Since $M$ is Noetherian, there exists $n$ such that $K_n = K_{n+1} = \cdots$. Let $x \in K_1$. Since $f$ is surjective, there exist $x_2, \dots, x_n$ such that
$x = f(x_2)$
$x_2 = f(x_3)$
$\dots$
$x_{n-1} = f(x_n)$
$x_n = f(x_{n+1})$
Since $x_{n+1} \in K_{n+1}$, $x_{n+1} \in K_n$. Hence $f^n(x_{n+1}) = 0$. Hence $x = f(x_2) = f^2(x_3) = \cdots = f^n(x_{n+1}) = 0$. QED
Theorem (a generalization of the theorem of Vasconcelos). Let $A$ be a commutative ring. Let $M$ be a finitely generated $A$-module. Let $N$ be an $A$-submodule of $M$. Let $f\colon N \rightarrow M$ be a surjective $A$-homomorphism. Then $f$ is injective.
Proof Let $0 \neq y_0 \in N$. It suffices to prove $f(y_0) \neq 0$. Let $f(y_0) = x_0$.
Let $x_1, \dots, x_n$ be generators for $M$. Let $f(y_i) = x_i$, $i = 1,\dots, n$.
Suppose $f(x_i) = \sum_{j = 1}^{n} a_{i, j} x_j, i = 0, 1,\dots, n$ and $y_i = \sum_{j = 1}^{n} b_{i, j} x_j, i = 0, 1,\dots, n$.
Let $B = \mathbb{Z}[a_{i, j}, b_{i, j}]$. $B$ is a Noetherian subring of $A$.
Let $P = Bx_1 + \cdots + Bx_n$, $Q = By_0 + By_1 + \cdots + By_n$. Since $y_i \in P, i = 0, 1, ..., n, Q \subset P$. Since $f(y_i) = \sum_{j=1}^{n} b_{i, j} f(x_j) \in P, f(Q) \subset P$. Hence $f$ induces a $B$-homomorphism $g\colon Q \rightarrow P$. Since $f(y_i) = x_i, i = 1,\dots, n$, $g$ is surjective. Hence, by the lemma, $g$ is injective. Hence $f(y_0) = g(y_0) \neq 0$ as desired. QED
Here is a quick and easy proof which I have found at the stacks project: If $M$ is cyclic, then $M \cong A/I$ for some ideal $I$. Then replace $A$ by $A/I$ so that wlog $M = A$. But then it's easy. In general, we do induction, but first we do the trick mentioned in Georges Elencwajg's answer: We endow $M$ with an $A[X]$-module structure such that multiplication with $X$ is the given endomorphism $T$. Thereby, we may assume that $T$ is multiplication with an element $X$ of the base ring $A$ - this means that every submodule is stable under this endomorphism(!). So we just pick some generator, look at its generated submodule $M'$ and apply the Five Lemma to $$\begin{array}{c} 0 & \rightarrow & M' & \rightarrow & M & \rightarrow & M/M' & \rightarrow & 0 \\ & & X\downarrow ~~~~&& X\downarrow ~~~~ && X \downarrow ~~~~ & & \\ 0 & \rightarrow & M' & \rightarrow & M & \rightarrow & M/M' & \rightarrow & 0\end{array}$$ and we are done.