Having already secured qualification for next year's tournament in Qatar, Germany smashed nine past Group J's bottom side to hand Flick a record-breaking sixth successive win since taking the reins in August.
The previous best start by a Germany coach was made by Flick's long-serving predecessor Joachim Loew, who won his first five games before a draw against Cyprus in 2006.
Loew, who stepped down after 15 years in charge earlier this year, was given a brief official send-off ahead of Thursday's game before watching Flick steal his record on a rainy night in Wolfsburg.
Germany began without several key players after defender Niklas Suele tested positive for coronavirus earlier in the week and Joshua Kimmich and Jamal Musiala were sent into quarantine as a precaution.
Yet the depleted hosts were able to cruise as the visitors imploded early in the first half.
Liechtenstein's Jens Hofer was sent off after just nine minutes when he caught Leon Goretzka in the neck with a high foot.
Ilkay Gundogan converted the resulting penalty, before Germany scored three goals in four minutes to make it 4-0 by the 25th minute.
Sane then set up Marco Reus with a tap-in moments later, and added another shortly after the break before another flurry of goals in the last 15 minutes.
Mueller added another five minutes from time, and the rout ended with another own goal as Maximilian Goeppel deflected a Baku cross into his own net.
Standout matches today