This Letter reports a measurement of the exclusive γγ → ℓ⁺ℓ⁻ (ℓ = e, μ) cross-section in proton-proton collisions at a centre-of-mass energy of 7 TeV by the ATLAS experiment at the LHC, based on an integrated luminosity of 4.6 fb⁻¹. For the electron or muon pairs satisfying exclusive selection criteria, a fit to the dilepton acoplanarity distribution is used to extract the fiducial cross-sections. The cross-section in the electron channel is determined to be σ(^(excl) / γγ →e⁺e⁻) = 0.428 ± 0.035 (stat.) ± 0.018 (syst.) pb for a phase-space region with invariant mass of the electron pairs greater than 24 GeV, in which both electrons have transverse momentum p(T) > 12 GeV and pseudorapidity |η| < 2.4. For muon pairs with invariant mass greater than 20 GeV, muon transverse momentum p(T) > 10 GeV and pseudorapidity |η| < 2.4, the cross section is determined to be σ(^(excl) / γγ → μ⁺μ⁻) = 0.628 ± 0.032(stat.) ± 0.021 (syst.) pb. When proton absorptive effects due to the finite size of the proton are taken into account in the theory calculation, the measured cross-sections are found to be consistent with the theory prediction.