Semileptonic kaon decays (Kl3 = K^+ → π^0 l^+ ν with l = e, μ) provide a clean way to extract the parameter |Vus| of the Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix. It is one of the fundamental parameters of the Standard Model and is only accessible via experimental measurement. The NA62 collaboration collected K^+ decay data in 2007 which are analyzed in the presented work. In total 28·10^6 Kμ3 , 37·10^6 Ke3 , and 2·10^6 K2π candidates were extracted from the data set and used to compute the ratios of branching fractions Br(Kμ3)/Br(K2π) = 0.16066(12)stat. (66)syst. = 0.1607(7), Br(Ke3)/Br(K2π) = 0.24320(17)stat. (41)syst. = 0.2432(4), and Br(Kμ3)/Br(Ke3) = 0.66061(17)stat. (245)syst. = 0.6606(25) which represent the most precise measurements available. The semileptonic branching fractions Br(Kμ3) = 3.321(19)% and Br(Ke3) = 5.027(22)% follow from the world average of Br(K2π) measurements. They are used to extract the CKM parameter |Vus| = 0.2219(10) . The new |Vus| result leads to the unitarity relation |Vud|^2 +|Vus|^2 +|Vub|^2 = 0.9983(6) which disagrees by 2.8 standard deviations with the CKM unitarity prediction. The e − μ lepton universality is tested with the ratio of the semileptonic branching fractions and agrees with the Standard Model prediction.