Abdullah Afzali, a member of Badghis provincial council, said security forces were still battling Taliban gunmen, and officials at the interior ministry in Kabul said that two of the three insurgents had been killed.
The Nangarhar attack came three days after Afghan forces killed several civilians, including a mother and her six children, during an air and ground raid in the northern Baghlan province on Tuesday.
Afghanistan's police and military personnel come under near-daily attacks and have suffered staggering casualties in recent months, most often at the hands of Taliban insurgents, who say the war will continue as long as USA and North Atlantic Treaty Organisation forces are still in Afghanistan.
Over 2,400 US service members have died since the war started in 2001, after the 9/11 attacks.
More than 2,400 American military personnel have been killed since the war began and more than 20,000 others have been wounded.
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The U.S. has been fighting the Taliban in Afghanistan since 2001.
The killings occurred a day after U.S. Secretary of State Mike Pompeo made a quick visit to the Afghan capital of Kabul where he said Washington was hopeful of a peace deal before September 1.
The Taliban insurgency said in a brief statement its fighters had planted a road side bomb in Sayedabad district of the eastern Afghan province of Wardak. The statement also said the identities of the soldiers would not be released.
The incident comes less than a week after Taliban officials were sitting face-to-face with United States negotiators in Doha in a bid to end Afghanistan´s almost 18-year-old war.
Indian security agencies said the terrorist cadre's shift to across the Durand Line has been done to avoid black-listing of Pakistan by the Financial Action Task Force (FATF) in its Paris conference later this year.
Pompeo added that discussions with the Taliban have also begun on US troop withdrawal.