Searches for exclusively produced W boson pairs in the process pp(γγ)→pW+W−p and an exclusively produced Higgs boson in the process pp(gg)→pHp have been performed using e±μ∓ final states. These measurements use 20.2 fb−1 of pp collisions collected by the ATLAS experiment at a center-of-mass energy s√=8 TeV at the LHC. Exclusive production of W+W− consistent with the Standard Model prediction is found with 3.0σ significance. The exclusive W+W− production cross section is determined to be σ(γγ→W+W−→e±μ∓X)=6.9±2.2(stat)±1.4(sys) fb, in agreement with the Standard Model prediction. Limits on anomalous quartic gauge couplings are set at 95% confidence level as −1.7×10−6<aW0/Λ2<1.7×10−6 GeV−2 and −6.4×10−6<aWC/Λ2<6.3×10−6 GeV−2. A 95% confidence-level upper limit on the total production cross section for an exclusive Higgs boson is set to 1.2 pb.