We present a detailed investigation of the $gamma$-ray emission in the vicinity of the supernova remnant (SNR) W28 (G6.4$-$0.1) observed by the Large Area Telescope (LAT) onboard the Fermi Gamma-ray Space Telescope. We detected significant $gamma$-ray emission spatially coincident with TeV sources HESS J1800$-$240A, B, and C, located outside the radio boundary of the SNR. Their spectra in the 2-100 GeV band are consistent with the extrapolation of the power-law spectra of the TeV sources. We also identified a new source of GeV emission, dubbed Source W, which lies outside the boundary of TeV sources and coincides with radio emission from the western part of W28. All of the GeV $gamma$-ray sources overlap with molecular clouds in the velocity range from 0 to 20 km s$^{-1}$. Under the assumption that the $gamma$-ray emission towards HESS J1800-240A, B, and C comes from $pi^0$ decay due to the interaction between the molecular clouds and cosmic rays (CRs) escaping from W28, they can be naturally explained by a single model in which the CR diffusion coefficient is smaller than the theoretical expectation in the interstellar space. The total energy of the CRs escaping from W28 is constrained through the same modeling to be larger than $sim$ 2 $times$ 10$^{49}$ erg. The emission from Source W can also be explained with the same CR escape scenario.
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