Hansi Flick deployed two forwards in the starting line-up for the first time during his tenure as head coach, and that decision reaped dividends in the 13th minute when the hosts took the lead thanks to Fullkrug.
Kai Havertz brought Nico Schlotterbeck's long ball under control for the Werder Bremen forward to sweep home a finish past Pedro Gallese - his fourth Germany goal in only his fifth cap.
Flick’s side looked in the mood to score a hatful, as Gallese then made an excellent double save to keep out strikes from Florian Wirtz and Timo Werner in quick succession.
A second did come through man-of-the-moment Fullkrug, who fired home a clinical first-time finish with debutant Marius Wolf providing a sumptuous ball into his path to make it five in five for his country.
Peru just about kept the deficit at two when Luis Advincula did well to clear away another Fullkrug effort and deny him a hat-trick shortly before the interval.
Juan Reynoso's side came out with renewed vigour after the break and had a penalty appeal dismissed when Fullkrug was judged not to have handled the ball from Renato Tapia's header.
But another chance fell Germany’s way when substitute Serge Gnabry hit the bar with an outstanding acrobatic effort from Joshua Kimmich’s delicious delivery into the area.
Flick’s men had a chance to extend their lead after being awarded a penalty following the intervention of VAR, as referee Maria Sole Caputi pointed to the spot upon seeing Schlotterbeck sent tumbling by Miguel Araujo.
However, Havertz spurned the opportunity and saw his subsequent spot-kick hit the post. The Chelsea man buried the rebound, but with no other player touching the ball, the goal did not stand.
Nevertheless, it did not dampen an otherwise fine night’s work for Die Mannschaft, who showed signs of recovery from a dismal showing at last year’s World Cup with a professional display in Mainz.
Meanwhile, the Incas showed plenty of endeavour, but lacked the quality to cause their hosts any significant problems.
Flashscore Man of the Match: Niclas Fullkrug (Germany)