Tottenham Hotspur playmaker Christian Eriksen had an otherwise quiet outing but provided the decisive moment when he threaded an immaculate pass through the defence for Poulsen to beat advancing Peru goalkeeper Pedro Gallese in the 59th minute.
RB Leipzig's exciting winger, Poulsen was fortunate not to be punished for a clumsy tackle inside the box as Cueva missed the first-half penalty.
In response Peru bought on their talisman and national team captain Paolo Guerrero, and he almost scored with a glorious backheel that had Schmeichel beaten.
Their colorful number have been a ubiquitous presence in Moscow in recent days, and they made their way en masse to Saransk - which has a population of just under 300,000 - to attend the team's World Cup Group C opening game against Denmark on Saturday. We're good defensively, which is important in football.
"At half-time we told him that he had to keep playing strong", Gareca told reporters.
Denmark looked more assured after the restart.
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The referee pointed to the spot upon reviewing video footage and Cueva blazed over, much to the dismay of Peru's sizable travelling support.
Denmark midfielder William Kvist was taken off on a stretcher after a sustaining a blow to the ribs in a challenge with Farfan, and disaster nearly struck for Age Hareide's men when Cueva was brought down in the penalty area by Poulsen.
After being awarded a penalty via VAR, the 26-year-old skied his effort from the spot and was consoled by his team-mates as they walked off at the end of the half.
Peru hopes to make a splash in its first appearance on the sports world's biggest stage in 36 years.
"They got very little and probably deserved more - sometimes that happens".
They also gathered around the city's main cathedral, playing music and bringing a touch of the South American party spirit.