The resonator transients and, in particular, the phenomenon of relaxation oscillations of diode end-pumped solid-state Cr:LiSAF and Nd:YVO4 lasers have been investigated at 850 nm and 1064 nm, respectively. The obtained results allow to understand clearly the resonator transients and show a generation of single laser pulses from the Cr:LiSAF and Nd:YVO4 lasers at near-threshold pumping levels. By proper choices of solid-state laser resonator and pumping parameters, obtainable duration of single solid-state laser pulses is computationally studied to be in nanosecond range and much shorter than the diode laser pumping pulse (100 μs). As a result, a technique of single short laser pulse generation at high repetition rate of diode pumping pulses has been experimentally demonstrated. In our Nd:YVO4 laser pumped by 100 μs diode laser pulse, single laser pulses as short as 93 ns at 1064 nm and pulse repetition rate as high as 3 kHz has been produced. The superior limit (about 3 kHz) in repetition rate of single Nd:YVO4 laser pulse mainly resulted from the fluorescence lifetime of active ion Nd3+.