The production cross-sections for $W\pm$ and $Z$ bosons are measured using ATLAS data corresponding to an integrated luminosity of 4.0 pb$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$ collected at a centre-of-mass energy $s=2.76$ TeV. The decay channels $W\to \ell \nu$ and $Z \rightarrow \ell \ell $ are used, where $\ell$ can be an electron or a muon. The cross-sections are presented for a fiducial region defined by the detector acceptance and are also extrapolated to the full phase space for the total inclusive production cross-section. The combined (average) total inclusive cross-sections for the electron and muon channels are: \begin{eqnarray} \sigma^{\text{tot}}{W^{+}\rightarrow \ell \nu}& = & 2312 \pm 26\ (\text{stat.})\ \pm 27\ (\text{syst.}) \pm 72\ (\text{lumi.}) \pm 30
(\text{extr.})\text{pb} \nonumber, \ \sigma^{\text{tot}}{W^{-}\rightarrow \ell \nu}& = & 1399 \pm 21\ (\text{stat.})\ \pm 17\ (\text{syst.}) \pm 43
(\text{lumi.}) \pm 21\ (\text{extr.})\text{pb} \nonumber, \ \sigma^{\text{tot}}_{Z \rightarrow \ell \ell}& = & 323.4 \pm 9.8
(\text{stat.}) \pm 5.0\ (\text{syst.}) \pm 10.0\ (\text{lumi.}) \pm 5.5 (\text{extr.}) \text{pb} \nonumber. \end{eqnarray} Measured ratios and asymmetries constructed using these cross-sections are also presented. These observables benefit from full or partial cancellation of many systematic uncertainties that are correlated between the different measurements.