Measurements of dijet $p_{\mathrm{T}}$ correlations in Pb+Pb and $pp$ collisions at a nucleon--nucleon centre-of-mass energy of $\sqrt{s_{\mathrm{NN}}}=2.76\textrm{ Te\kern -0.1em V}$ are presented. The measurements are performed with the ATLAS detector at the Large Hadron Collider using Pb+Pb and $pp$ data samples corresponding to integrated luminosities of 0.14 nb$^{-1}$ and 4.0 pb$^{-1}$, respectively. Jets are reconstructed using the anti-$k_t$ algorithm with radius parameter values $R=0.3$ and $R=0.4$. A background subtraction procedure is applied to correct the jets for the large underlying event present in Pb+Pb collisions. The leading and sub-leading jet transverse momenta are denoted $p_{\mathrm{T_{\mathrm{1}}}}$ and $p_{\mathrm{T_{\mathrm{2}}}}$. An unfolding procedure is applied to the two-dimensional ($p_{\mathrm{T_{\mathrm{1}}}}$, $p_{\mathrm{T_{\mathrm{2}}}}$) distributions to account for experimental effects in the measurement of both jets. Distributions of $(1/N)\mbox{$\mathrm{d}$} N/\mbox{$\mathrm{d}$} x_{\mathrm{J}}$, where $x_{\mathrm{J}}=p_{\mathrm{T}_{2}}/p_{\mathrm{T}_{1}}$, are presented as a function of $p_{\mathrm{T_{\mathrm{1}}}}$ and collision centrality. The distributions are found to be similar in peripheral Pb+Pb collisions and $pp$ collisions, but highly modified in central Pb+Pb collisions. Similar features are present in both the $R=0.3$ and $R=0.4$ results, indicating that the effects of the underlying event are properly accounted for in the measurement. The results are qualitatively consistent with expectations from partonic energy loss models.