The Manchester City midfielder gave the hosts an 11th minute lead from the penalty spot, sending the goalkeeper the wrong way after Jens Hofer was dismissed for recklessly kicking Leon Goretzka in the neck in an attempt to clear a cross.
It was one way traffic after that, with a Leroy Sane brace, a Marco Reus strike and a Daniel Kaufmann own goal putting Germany four goals to the good before Gundogan was withdrawn in the 64th minute.
Two efforts from Thomas Muller, one from Ridle Baku and another own goal from Maximilian Göppel completed the rout in a contest in which the already qualified home side registered no fewer than 42 shots.
In European Under-21 Championship Qualifying, Cole Palmer provided an assist before being forced off with a suspected injury as England recorded an important 3-1 victory over the Czech Republic.
The Academy graduate picked out Anthony Gordon, who opened the scoring on his debut with an 18-yard effort that beat the visiting goalkeeper thanks to a deflection off centre-back Martin Vitik.
It was the third goal Palmer has been directly involved in from his four appearances for the Under-21s, though his night ended prematurely when he limped off in the 73rd minute.
The 19-year-old midfielder started alongside the on-loan Taylor Harwood-Bellis at Turf Moor, where fellow loanee Tommy Doyle was an unused substitute, as Lee Carsley’s side deservedly took all three points.
Gordon added his and England's second in the 11th minute, before Arsenal’s Folarin Balogun netted a third on the half hour mark.
The Czech Republic pulled one back when Adam Karabec converted their only shot on target from the penalty spot after he had been fouled by Luke Thomas.
The hosts produced a professional second half display to see out the game and they move up to second in Group G, two points behind the Czech Republic, but with a game in hand.
Pablo Sarabia’s strike was enough for Luis Enrique’s side to leapfrog Sweden into top spot ahead of their final game against the Scandinavians, which will determine who is heading to next winter’s tournament and who must enter the playoffs.
Oleksandr Zinchenko was also in action as Ukraine draw 1-1 with Bulgaria in a friendly.
The 24-year-old played the entire game as Teras Stepanenko’s stunning strike rescued a point for the hosts after Radoslav Kirilov had put Bulgaria in front in the first half.
Ruben Dias and Joao Cancelo were unused substitutes in Portugal’s 0-0 draw with the Republic of Ireland in Group A of World Cup qualifying.
With a crunch game against Serbia to follow, the City duo were left out by Fernando Santos, who had Pepe sent off as his side were frustrated in Dublin.
Santos reported the playmaker was suffering with a muscle issue ahead of the game, but he is expected to return for their final match against Serbia, which will determine who finishes top of the group to seal automatic qualification to Qatar 2022.