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After a first half of few chances it was visitors Armenia who took the lead in Copenhagen when Borussia Dortmund's Henrikh Mkhitaryan found the bottom corner with a low left-foot drive from 20 yards.
Fifteen minutes later Denmark were level when Bayern Munich youngster Pierre Hojbjerg fired home from outside the area for his first senior international goal.
It also made him the youngest scorer for Denmark in Euro qualifiers, breaking the previous record of Tottenham Hotspur's Christian Eriksen.
Both sides had chances to net the third and it was Denmark who scored the vital goal with 10 minutes on the clock.