The result you're looking for is contained in the following article :

Haberland, Klaus. Perioden von Modulformen einer Variabler and Gruppencohomologie I (German)  [Periods of modular forms of one variable and group cohomology I], Math. Nachr.  112  (1983), 245-282.

Let $S_k$ (resp. $M_k$) be the space of holomorphic cusp forms (resp. holomorphic modular forms) for $\Gamma = SL_2(\mathbf{Z})$. Let $\Gamma_{\infty}$ be the stabilizer of $\infty$ in $\Gamma$. Let $V_k$ be the space of polynomials of degree $\leq k-2$ with complex coefficients. Haberland proves an exact sequence

\begin{equation}
(*) \qquad  0 \to S_k \oplus \overline{S_k} \to H^1(\Gamma,V_k) \to H^1(\Gamma_\infty,V_k) \to 0.
\end{equation}
Let $T =\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} \in \Gamma_{\infty}$. There is a natural map $V_{k+1} \to H^1(\Gamma_\infty,V_k)$ sending a polynomial $P$ to the cocycle $c_P$ determined by $c_P(T) = P(X+1)-P(X)$. It is easy to check that this map induces an isomorphism $\psi : V_{k+1}/V_k \cong H^1(\Gamma_\infty,V_k)$, so that the latter space is one-dimensional.

The "Eisenstein cocycle" you're looking for is a natural map $\delta : M_k \to H^1(\Gamma,V_k)$ which Haberland constructs the following way (actually I learnt this construction and many other properties of $\delta$ during Zagier's 2002-2003 lectures at the Collège de France).

Let $f \in M_k$. Let $\widetilde{f}$ be an Eichler integral of $f$, that is any holomorphic function on $\mathcal{H}$ such that

\begin{equation}
\bigl(\frac{1}{2\pi i} \frac{d}{dz}\bigr)^{k-1} \widetilde{f}(z) = f(z).
\end{equation}
Note that $\widetilde{f}$ is unique up to adding some element of $V_k$.

For any $n \in \mathbf{Z}$, let $|_n$ denote the weight $n$ action of $\Gamma$ on the space of complex-valued functions on $\mathcal{H}$ (so that any $f \in M_k$ is a fixed vector of the weight $k$ action). The crucial fact is that we have

\begin{equation}
\widetilde{f} |_{2-k} (\gamma-1) \in V_k \qquad (\gamma \in \Gamma).
\end{equation}
This can be proved using Bol's identity

\begin{equation}
(\frac{d}{dz})^{k-1} (F |_{2-k} g) = (\frac{d^{k-1} F}{dz^{k-1}}) |_k g
\end{equation}
which holds for any holomorphic function $F$ on $\mathcal{H}$ and any $g \in SL_2(\mathbf{R})$.

Since $\gamma \mapsto \widetilde{f} |_{2-k} (\gamma-1)$ is obviously a coboundary in the space of functions on $\mathcal{H}$, it defines a cocycle $r_f$ in the space $V_k$, and therefore an element of $H^1(\Gamma,V_k)$. Thus we have constructed $\delta : M_k \to H^1(\Gamma,V_k)$. This map is compatible with the Eichler-Shimura isomorphism

It is not difficult to check that if $f =\sum_{n \geq 0} a_n e^{2i\pi nz}$ then the image of $\delta(f)$ in $H^1(\Gamma_\infty,V_k)$ is the image of the polynomial $\frac{a_0}{k-1} \cdot X^{k-1} \in V_{k+1}$ under the isomorphism $\psi$ above. In particular $\delta$ is injective, and the exact sequence $(*)$ gives the isomorphism you want.

Note : there is also a formula for the value of the cocycle $r_f$ at the matrix $S=\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$, in terms of the special values $L(f,s)$ for $1 \leq s \leq k-1$, which was also explained in Zagier's lectures.

Please tell me if something isn't clear in my explanation.