The gunmen carried out three separate shootings in two Shiite towns on Saturday, provincial police spokesman Ziad al-Rukaiti said.
"The prison in Qatif came under fire from an unknown source," said Rukaiti, referring to the prison in the Shiite town of Awamiya in the Qatif district.
Rukaiti said the shots directed at the prison were "followed by shooting at a police checkpoint" in Awamiya, and another shooting targeting a police patrol in a separate Shiite town.
Rights activists confirmed that an 18-year-old woman and her two-year-old child "were wounded by random gunfire in Awamiya."
Since early 2011, mainly Shiite towns in Eastern Province have seen sporadic protests and confrontations between police and Shiite residents who complain of marginalization.
There are an estimated two million Shiites in the Sunni-dominated kingdom of around 27.5 million people.