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}
\left(\frac{1}{2\pi i} \frac{d}{dz}\right)^{k-1} \widetilde{f}(z) = f(z).
\end{equation}
Note that $\widetilde{f}$ is unique up to adding some element of $V_k$. Since we integrate $k-1$ times, the function $\widetilde{f}$ should be thought of as a function of "weight" $k-2\cdot (k-1) = 2-k$ (of course this isn't true in the strict sense). Let us make this more precise.

For any $n \in \mathbf{Z}$, let $|_n$ denote the weight $n$ action of $SL_2(\mathbf{R})$ 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 of $\Gamma$). Note also the weight $2-k$ action gives the usual action of $\Gamma$ on $V_k$. 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}
\left(\frac{d}{dz} \right)^{k-1} (F |_{2-k} g) = \left(\frac{d^{k-1} F}{dz^{k-1}} \right) |_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 in the space $V_k$. Therefore we get $\delta(f) \in H^1(\Gamma,V_k)$ and this element doesn't depend on the choice of $\widetilde{f}$. Thus we have constructed $\delta : M_k \to H^1(\Gamma,V_k)$.

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 $\frac{a_0 \cdot (2\pi i)^{k-1}}{(k-1)!} \cdot X^{k-1} \in V_{k+1}$ under the isomorphism $\psi$ above. In particular $\delta$ is injective, and the exact sequence $(*)$ gives the isomorphism you want.

Note that there is a distinguished choice of $\widetilde{f}$, namely

\begin{equation}
\widetilde{f} = \sum_{n \geq 1} \frac{a_n}{n^{k-1}} e^{2i\pi nz} + \frac{a_0 \cdot (2\pi i)^{k-1}}{(k-1)!} z^{k-1}.
\end{equation}
Let $c_f \in Z^1(\Gamma,V_k)$ be the cocycle associated to this choice of $\widetilde{f}$. Let us compute the value of $c_f$ on $T$ and $S= \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$. First as explained above, we have

\begin{equation}
c_f(T)=\frac{a_0 \cdot (2\pi i)^{k-1}}{(k-1)!} ((X+1)^{k-1}-X^{k-1}).
\end{equation}
To compute $c_f(S)$, Haberland uses the natural integral representation of $\widetilde{f}$ in terms of $f-a_0$, and gets

\begin{equation}
c_f(S) = \frac{(2\pi i)^{k-1}}{(k-2)!} \int_0^{\infty} \left(f(z)-\frac{a_0}{z^k}-a_0 \right) (z-X)^{k-2} dz
\end{equation}
(there is a similar but more complicated formula for $c_f(\gamma)$ for any $\gamma \in \Gamma$, see below). Then $c_f(S)$ can be expressed in terms of the special values of $L(f,s) := \sum_{n=1}^\infty a_n/n^s$ at integers $s = 1,\ldots,k-1$. It is then a good exercise to compute $c_f(S)$ when $f$ is the Eisenstein series $E_k$, in terms of Bernoulli numbers and of $\zeta(k-1)$ (this is Satz 3 in Haberland's article, Kapitel 1).

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

EDIT : I found the following expression for $c_f(\gamma)$ where $\gamma \in \Gamma$. It is quite complicated (maybe it could be somewhat simplified) :

\begin{equation}
\begin{aligned}
\frac{(k-2)!}{(2\pi i)^{k-1}} c_f(\gamma) &= \int_{z_0}^{\infty} (f(z)-a_0)(z-X)^{k-2} dz + \int_{\gamma^{-1} \infty}^{z_0} \left(f(z) -\frac{a_0}{(cz+d)^k} \right) (z-X)^{k-2} dz \\
& + \frac{a_0}{k-1} \left((X-z_0)^{k-1}-(X-\gamma z_0)^{k-1} |_{2-k} \gamma + X^{k-1} |_{2-k} (\gamma-1) \right)
\end{aligned}
\end{equation}
where $z_0 \in \mathcal{H}$ is arbitrary and $\gamma= \begin{pmatrix} a & b \\ c & d \end{pmatrix}$.