A double from Timo Werner, sandwiched between second half goals for Chelsea teammate Kai Havertz and starlet Jamal Musiala, secured Hansi Flick’s side a resounding win in Skopje.
It leaves them safely through from UEFA qualifying Group J with an eight-point lead over Romania with two games to play.
Elsewhere in European qualifying action, Wales took a vital step forward as they look to secure second in Group E with a nervy 1-0 win over Estonia in Tallinn.
Kiefer Moore’s early goal proved the difference between the two sides as the hosts squandered a couple of chances to draw level.
It leaves Wales’ fate in their own hands, level on points with the Czech Republic but with a game in hand, with Belgium likely to top of the group.
However striker Moore will now miss Wales’ next game against Belarus after the referee deemed a raised arm to be worthy of a yellow card as he jostled for position in the box.
“It is disappointing – I want to play every game,” Moore told Sky Sports after full-time.