In this paper, the author derives new proofs of Eisenstein series identities associated with the Borweins functions \(a(q)\), \(b(q)\) and \(c(q)\), defined by \[ \begin{aligned} a(q) &= \sum_{m,n=-\infty}^\infty q^{m^2+mn+n^2},\\ b(q) &= \sum_{m,n=-\infty}^\infty e^{2(m-n)\pi i/3}q^{m^2+mn+n^2},\\ \text{and} c(q) &=\sum_{m,n=-\infty}^\infty q^{(m+1/3)^2+(m+1/3)(n+1/3)+(n+1/3)^2}.\end{aligned} \] Most of these identities are new and some of them are truly elegant. The proofs of these identities involve the use of Residue Theorem on cleverly chosen functions. One of the identities proved by the author, namely, \[ 1+6\sum_{n=1}^\infty \left\{ \frac{nq^n}{1-q^n} - \frac{5nq^{5n}}{1-q^{5n}}\right\} =a(q)a(q^5)+2c(q)c(q^5), \] is recently employed to derive the following new series for \(1/\pi\): \[ \frac{4\sqrt{3}}{\pi\sqrt{5}} = \sum_{k=0}^\infty (9k+1)\frac{\left(\frac{1}{3}\right)_k\left(\frac{2}{3}\right)_k \left(\frac{1}{2}\right)_k}{(k!)^3}\left(-\frac{1}{80}\right)^k. \] The paper ends with a proof of the Borweins' cubic inversion formula, which expresses \(a(q)\) in terms of \(c^3(q)/a^3(q)\).