The military said that during the operations, the soldiers came under heavy small-arms fire and rocket-propelled grenade attacks in both places.
"Using appropriate escalation of force measures, ground forces were forced to use close air support to suppress the enemy fire. During the close air support, at least 20 terrorists were killed," it said.
Latif al-Tamimi, chief of the security committee on the Maysan provincial council, called the operation a "catastrophe", accusing troops of firing randomly.
Iraqi police said the al-Mahdi Army, the fighters loyal to Muqtada al-Sadr, a populist Shia leader, was allegedly involved in the clashes, which lasted for about two hours before dawn.
Elsewhere on Monday, two people were killed in clashes between Iraqi police and al-Mahdi Army in Nasiriya, about 120km south of Amara, police said.
The fighting began after a number of police patrols were attacked on Sunday night, a police officer and an official in the town's health department said.
British forces handed security control of Maysan province back to Iraqi security forces in April.