The spokesman was quoted as saying by the Almasirah news channel that Saturday night attacks aimed at "sensitive sites" in the capital as well as at military and economic targets in Jazan, Najran and Asir.
The attacks were launched in retaliation for air strikes by the Saudi-led military coalition on Houthi positions in Yemen's capital Sana and areas close to the southern Saudi border.
Cross-border violence comes despite a call by the UN chief on Thursday to cease fire amid the deadly coronavirus pandemic. The internationally recognised Yemeni government and its Saudi supporters reportedly backed the proposal as did the Houthis.
A Saudi-led coalition of mostly Arab states has conducted military operations in support of Yemeni President Abdrabuh Mansour Hadi against the Houthi movement since 2015.