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.