They said the unrest in the province of Qatif late on Monday had been incited by "a foreign country", without elaborating.
Saudi Arabia’s minority Shia population is concentrated in the east, the scene of protests earlier this year.
State media said eight of those wounded were security personnel and three were civilians.
State news agency SPA quoted the interior ministry as saying that "a group of outlaws and rioters on motorbikes" had gathered in al-Awamia village near the city of Qatif, "carrying petrol bombs".
The group was responsible for acts leading to "insecurity with incitement from a foreign country that aims to undermine the nation’s security and stability", SPA reported.
Saudi mentions of foreign meddling are normally veiled references to Iran, the region’s main Shia power, observers say.
In March, Saudi police opened fire to disperse protesters in Qatif, a day before planned countrywide anti-government protests.
The protesters, from the Shia minority, were demanding the release of prisoners they said had been held without charge.
Protests are illegal in Saudi Arabia, which has had an absolute monarchy since its unification in the 1930s.
Rights groups have accused the police of beating protesters during previous rallies in Qatif.
Shias make up about 10% of the population in Sunni-dominated Saudi Arabia.
Saudi Arabia has not seen protests on the same scale as other nations in the Middle East and North Africa during the so-called Arab Spring.