A single-shot damage induction to plasmid DNA was demonstrated by applying X-rays emitted by a laser-produced plasma. Yields of single-strand breaks and double-strand breaks were determined as a function of energy fluence which was adjusted by varying the distance of the exposed sample from the X-ray source and by thickness of Al filters attenuating X-rays. As an intense source of X-ray radiation was employed a double-stream gas puff target irradiated by sub-kJ, near-infrared (NIR) focused laser pulses at the PALS facility (Prague Asterix Laser System) to produce high-energy pulses of soft X-rays from hot, dense Xe plasma. The double-puff arrangement ensures high gas density and conversion efficiency from NIR to X-rays approaching that typical for solid targets. In addition, its major advantage over solid targets is that it is debris free and has substantially suppressed charged particle emission.