Denmark did not get its first shot on goal until the 39th minute, when Eriksen drove a direct kick into a wall of defenders, and Lasse Schonne volleyed the rebound right at Gallese.
Peru ultimately could not find an equaliser and Denmark joined France on three points atop the group. Peru sit third, above Australia, and have their work cut out if they are to progress to the knockout stages.
After the game, Hareide was magnanimous in victory over a Peru side who were taking part in their first World Cup game for 36 years. The star striker was ruled out of the World Cup for doping but earned permission to play, cutting his ban short, just prior to the tournament.
Ricardo Gareca's pacy side took the game to Denmark and were certainly not afraid to shoot on sight. However, the forward stuttered in his run up and then blasted an bad effort way over the crossbar.
The referee initially waved Peru's appeals away after Cueva appeared to be tripped by Poulsen but as play continued, the decision was being reviewed and the official eventually halted the action to point to the spot.
Within the opening minutes of the match, Peru was pushing forward and almost drew a penalty when Flores and Poulsen collided on the left side of the area, but Gassama waved play on.
Peru will feel hard done by at the result after creating far more chances than Denmark, but neither team can have any complaints about VAR's involvement.