Iraq's Interior Ministry confirmed the attacks on Sunni mosques late Sunday in Hilla, around 100 km (60 miles) south of Baghdad. Prime Minister Haider al-Abadi blamed them on "Daesh (Islamic State) and those who are similar to them," without further explanation.
He ordered provincial authorities "to chase the criminal gangs" who attacked the mosques.
Iraq has faced sectarian bloodletting for years, mainly between minority Sunnis and a Shi'ite majority empowered after the US-led invasion in 2003.
The battle against Sunni Islamic State militants who control large swathes of the north and west has only exacerbated those tensions.
The spark for Sunday's attacks appears to have been Nimr's execution a day earlier, which triggered angry reactions in Shi'ite-ruled Iraq and Iran.
Saudi Arabia cut ties with regional rival Iran on Sunday after protesters attacked the kingdom's embassy in Tehran. Bahrain, the Shi'ite-majority Gulf state ruled by a Sunni family, and Sudan followed suit on Monday.
The attack on a mosque in central Hilla destroyed its dome and several walls, according to a Reuters TV cameraman who visited the site. Provincial council member Falah al-Khafaji and a police source said a guard in the building was killed.
"We saw smoke rising from the dome of the mosque. We found all the walls destroyed and the furniture inside in shambles," said resident Uday Hassan Ali.
Another mosque in Hilla's northern outskirts was also attacked and a Sunni cleric was killed in a separate incident in Iskandariya, about 40 km (25 miles) south of Baghdad, Khafaji and the police source said.
"We have leads and security measures will be taken near mosques," said Khafaji, pledging to rebuild the buildings.
Prominent religious and political leaders in Iraq have called on the government to cut ties with Saudi Arabia, which reopened its Baghdad embassy last week after closing it in 1990 following the Iraqi invasion of Kuwait.
While Abadi and Iraq's foreign ministry have condemned Nimr's execution, they have given no indication of a more severe response.
Thousands of protesters marched in Baghdad and Shi'ite cities in southern Iraq, heeding calls by prominent Shi'ite cleric Moqtada al-Sadr to protest against al-Nimr's execution.
In Baghdad, demonstrators carrying portraits of Nimr rallied outside the heavily fortified Green Zone housing government departments and diplomatic representations, including the newly reopened Saudi embassy.
Police guarding the zone pushed back a group trying to cross a line of barbed wire as they chanted "damned, damned be Al Saud," referring to the Saudi ruling family.
Similar protests were held in Basra, southern Iraq's biggest city, and in the Shi'ite holy cities of Najaf and Kerbala.