In this work, a method is proposed and demonstrated for fabrication of chirped fiber Bragg gratings (CFBGs) in single-mode fiber by femtosecond laser point-by-point inscription. CFBGs with bandwidths from 2 to 12 nm and dispersion ranges from 14.2 to 85 ps/nm are designed and achieved. The sensitivities of temperature and strain are 14.91 pm/°C and 1.21pm/µε, respectively. Compared to the present phase mask method, femtosecond laser point-by-point inscription technology has the advantage of manufacturing CFBGs with different parameter flexibilities, and is expected to be widely applied in the future.