Here is an alternative take on Donu's argument: Removing the image of a section of $E_\Gamma$ allows one to regard a fiber of $E_{\Gamma}$ as a once-punctured torus $S$. (In order to construct a section, use the standard upper half-space model of the Teichmuller space of the tori, so that each marked torus is identified with the fundamental parallelogram $P$ with vertices $0, 1, \omega, \omega+1\in {\mathbb C}$. Then the section is given by the map $\omega\to 0\in P$.) Then $E_{\Gamma}$ is isomorphic to the quotient of the Teichmuller space $T(S)$ of $S$ by a finite-index torsion-free subgroup $Mod^o_S$ of the mapping class group $Mod_S$ of $S$. Now, this is a general fact (Deligne-Mumford, et al) that $T(S)/Mod^o_S$ is quasi-projective (for any Riemann surface of finite type). In the special case you are interested in, it seems that quasi-projectivity of $E_{\Gamma}$ was first proven by Kodaira (On compact analytic surfaces. II, III), at least, Shioda (On elliptic modular surfaces, 1972) attributes the result to him.

Here is an alternative take on Donu's argument: Removing the image $\sigma$ of a section of $E_\Gamma$ allows one to regard a fiber of $E_{\Gamma}$ as a once-punctured torus $S$. (In order to construct a section, use the standard upper half-space model of the Teichmuller space of the tori, so that each marked torus is identified with the fundamental parallelogram $P$ with vertices $0, 1, \omega, \omega+1\in {\mathbb C}$. Then the section is given by the map $\omega\to 0\in P$.) Then $M=E_{\Gamma}\setminus \sigma$ is isomorphic to the quotient of the Teichmuller space $T(S)$ of $S$ by a finite-index torsion-free subgroup $Mod^o_S$ of the mapping class group $Mod_S$ of $S$. Now, this is a general fact (Deligne-Mumford, et al) that $T(S)/Mod^o_S$ is quasi-projective (for any Riemann surface of finite type). It is not hard to see that DM compactification of $T(S)/Mod^o_S$ is our case will add (among other things) the curve $\sigma$ back to $M$, thus, providing a projective compactification of $M$. 

In the special case you are interested in, it seems that quasi-projectivity of $E_{\Gamma}$ was first proven by Kodaira (On compact analytic surfaces. II, III), at least, Shioda (On elliptic modular surfaces, 1972) attributes the result to him.

Addendum: As an alternative to this argument, one can use Ron Livne's thesis "On certain covers of the universal elliptic curve". Livne proves that for every level $N\ge 5$ congruence subgroup $\Gamma(N)$ in the modular group $SL(2, {\mathbb Z})$, the universal elliptic curve over $E(N):={\mathbb H}^2/\Gamma(N)$ admits a degree $d\ge 2$ cyclic branched cover $E_d(N)$, so that the latter admits a compactification $X_d(N)$ (compatible with branched cover), so that $X_d(N)$ is a general type projective surface. Thus, every $E_\Gamma$ admits a finite regular cover which is biholomorphic to a finite cover over one of the $E_d(N)$'s. Thus, $E_\Gamma$ is quasi-projective.

Livne also refers to Mumford's paper "Prym varieties. I." Contributions to analysis (a collection of papers dedicated to Lipman Bers), pp. 325–350. Academic Press, New York, 1974, for a direct proof of quasi-projectivity of $E(N)$'s. I do not have access to Mumford's paper, so I cannot say for sure.