The fighting, part of an escalation in violence between the two sides in recent months, came amid a security clampdown on the capital, Colombo, following a pair of rebel bombings that killed 20 people.
On Thursday, troops killed four guerrilla fighters in a battle near the rebels' stronghold in northern Sri Lanka , the military said Friday. Four others were killed in two separate battles in the region, the military said in a statement.
The Tamil Tigers have been fighting since 1983 to create a separate homeland for Sri Lanka's minority ethnic Tamils following decades of discrimination by governments controlled by the Sinhalese majority.
The group, branded a terror organization by the U.S. and European Union, is responsible for more than 240 suicide bombings as well as scores of other deadly attacks.
Rebel spokesman Rasiah Ilanthirayan, who has not answered repeated calls from The Associated Press since the latest bombings Wednesday, could not be reached for comment on the new fighting.
The death toll from the worst of the bombings - at a department store in a Colombo suburb - rose to 19 Thursday as two of the injured died from their wounds, security forces said. An earlier suicide bombing targeting a Cabinet minister killed one person in addition to the bomber.
U.N. Secretary-General Ban Ki-moon condemned the rebel attacks and called on both sides to return to peace talks. The U.S., in its condemnation of attacks on civilians, said "only a political solution, not a military one, offers a way out of the current cycle of escalating violence."
The European Union called on both sides to avoid civilian casualties.
Fearing the attacks Wednesday were only the first in a wave of bombings, the government canceled school for the rest of the week. It also cut train service from the northern town of Vavuniya to the capital in the belief that rebels were using the route to infiltrate Colombo.
The military, which already maintained a maze of checkpoints throughout the capital, sent more forces into the streets, and intelligence agencies were directed to redouble their efforts to prevent more attacks, government spokesman Keheliya Rambukwella said.
The co-author of this disaster is the Dutch government, which did not find either strength or desire to save the lives of its citizens who were flying on that plane. The Dutch authorities did not demand Ukraine to comply with international aviation regulations
On the second day of the St. Petersburg International Economic Forum, a plenary meeting was held, in which Russian President Vladimir Putin, French President Emmanuel Macron, Japanese Prime Minister Shinzo Abe, Chinese Vice President Wang Qishan and IMF head Christine Lagarde took part