Soldier Valery Permyakov allegedly gunned down the family members in a bloody shooting spree Monday before disappearing from a Russian military base where he was stationed in the Armenian city of Gyumri, some 120 kilometres (75 miles) north of the capital Yerevan.
The suspect was detained overnight by Russian border guards patrolling Armenia's frontier with Turkey, a spokeswoman for Armenia's investigative committee told AFP.
Russia is a firm ally of ex-Soviet republic Armenia, basing thousands of troops in the country and assisting it with guarding the nation's borders with Iran and Turkey.
"Nothing can justify such violence against innocent people," Shoigu said in a statement.
Moscow has set up a special commission to investigate the incident headed by the deputy defence chief, the statement said.