The clashes have also focused attention on the role of the United Nations, which is deploying a new force with a mandate to attack rebel groups in eastern Democratic Republic of Congo.
The 3,000-strong Intervention Brigade, made up of South African, Tanzania and Malawian troops, has begun patrols but not yet engaged in combat.
The United Nations has warned it would block any attack on Goma, a city of one million people bordering Rwanda, which was briefly captured by rebels in November.
A Reuters reporter in Mutaho, some 7 km (4 miles) northeast of Goma, saw three army attack helicopters bombard rebel positions in the town of Kibati, 4 km further north of Goma.
"The situation is under control," army spokesman Colonel Olivier Hamuli told Reuters in Mutaho. "We were attacked on Sunday and our troops are pushing the enemy forces back."
Rebels and government troops traded mortar fire on Monday close to the northern and western outskirts of Goma. The United Nations said that a shell fell on Tuesday 100 meters (yards) from Goma airport, with no victims reported.
"This recurrence of fighting close to inhabited areas poses a serious protection issue for thousands of people and could trigger some drastic humanitarian consequences," said Moustapha Soumare, U.N. humanitarian coordinator in Congo.
Millions of people have died from violence, disease and hunger since the 1990s as foreign-backed rebel groups have fought for control of eastern Congo's rich deposits of gold, diamonds, copper, cobalt and uranium, destabilizing the Great Lakes region at the heart of Africa.
Kinshasa repeated claims on Monday that Rwanda was directly backing the Tutsi-led M23 rebels. A U.N. report said the group recruits in Rwanda with the aid of sympathetic military officers.
Kigali, which has in the past backed insurgents in Congo, denies any support for M23. It accused Kinshasa and U.N. troops on Monday of "provocative and deliberate" shelling of its territory, though it said no-one was wounded.
ATTACK ON U.N. PATROL
The 17,000-strong U.N. force in Congo (MONUSCO), the world's largest peacekeeping mission, has been deployed for more than a decade but has failed to stem a conflict in which millions have died from violence, hunger and disease since the 1990s.
The arrival of the Intervention Brigade has raised hopes of peace. The World Bank is offering $1 billion to regional governments to promote development if they respect a U.N.-brokered February deal not to back rebels in mineral-rich eastern Congo.
Peace talks between the Congolese government and M23 in Kampala, the capital of neighboring Uganda, have stalled.
With the United Nations saying it would intervene with "lethal force" to protect civilians, the Congolese army has encouraged people displaced by the fighting to return home.
Underscoring the challenge of pacifying eastern Congo, MONUSCO said on Monday that one of its patrols had been ambushed by Ugandan ADF rebels near the town of Beni, some 250 km (150 miles) to the north of Goma.
Two U.N. vehicles were damaged in the attack which was fought off by Nepalese and Jordanian peacekeepers on their way to investigate reports of rights abuses by the ADF.
The Red Cross estimates that 66,000 Congolese have refugees fled into Uganda since Thursday after attacks by the ADF, an Islamist group which Kampala says is allied to elements of Somalia's al Shabaab movement, an al Qaeda-linked group.
(Additional reporting by Stephanie Ulmer-Nebehay in Geneva; Writing by Daniel Flynn; Editing by Angus MacSwan)