We present the first observation and cross section measurement of exclusive dijet production in pp interactions, pp→p+dijet+p. Using a data sample of 310  pb−1 collected by the Run II Collider Detector at Fermilab at s√=1.96  TeV, exclusive cross sections for events with two jets of transverse energy EjetT≥10  GeV have been measured as a function of minimum EjetT. The exclusive signal is extracted from fits to data distributions based on Monte Carlo simulations of expected dijet signal and background shapes. The simulated background distribution shapes are checked in a study of a largely independent data sample of 200  pb−1 of b-tagged jet events, where exclusive dijet production is expected to be suppressed by the Jz=0 total angular momentum selection rule. Results obtained are compared with theoretical expectations, and implications for exclusive Higgs boson production at the pp Large Hadron Collider at s√=14  TeV are discussed.