The ministry said Youssef al-Suleiman carried out the attack on the police compound in the city of Abha, the provincial capital of Asir, just after the Islamic State group released a still image of the man and an audio recording purportedly from him.
They identified the bomber as Abu Sinan al-Najdi and the audio included a warning that Saudi rulers and troops "will not enjoy peace" for taking part in the US-led coalition battling the Islamic State group in Iraq and Syria.
Hours after Thursday's bombing, a previously unheard of Islamic State affiliate, which calls itself Hijaz Province of the Islamic State, claimed responsibility for the attack.
The kingdom had for years quietly allowed thousands of Saudis to leave the country to join militant groups fighting in Iraq and Syria, until the late King Abdullah last year decreed that fighting abroad was illegal. A wave of Islamic State group attacks has hit the kingdom in recent months, many carried out by Saudi citizens.