Despite a minor tumble in the preparatory laps in the first leg of the final, Vogel powered, after a restart, to a 2-0 victory against Elis Ligtlee of the Netherlands.
Zhong Tianshi of China took the bronze.
Germany, with three golds, one silver and three bronze medals, lead hosts France who have three golds and one bronze, which they collected in the men's individual pursuit on Saturday.
Julien Morice finished third in the event to snatch France's first medal in the discipline since 1998.
Swiss Stefan Kueng won gold after Australian Jack Bobridge, who last month failed in his Hour Record attempt, cracked in the last of four kilometres to settle for silver.
The evening session was delayed by about 20 minutes as drops of water dripped from the roof onto the track, in an incident reminiscent of the London Olympics.
Earlier, Colombian Fabian Puerta crashed heavily during a training session and was taken to hospital "for further assessment" after being treated on site by the velodrome's medical team, the International Cycling Union (UCI) said.
Colombia's Fernando Gaviria claimed gold in the men's omnium ahead of Australian Glenn O'Shea with Italian Elia Viviani, who also rides on the road with Team Sky, taking the bronze.
Britain faces the prospect of not winning a world title for the first time since 2001 unless Olympic champion Laura Trott manages to turn things around on Sunday.
The 22-year-old lies third in the women's omnium after three of six events, following victories in the individual pursuit and the elimination race.
Her fiance and fellow Olympic champion Jason Kenny was knocked out in the men's sprint last 32.
"I just fluffed it up," Kenny said. "I just got caught napping.
The noise is expected to reach deafening levels in the arena on Sunday when defending champion Francois Pervis, who has already won two golds here, takes on fellow Frenchman Gregory Bauge, the 2010 and 2012 champion, in the sprint quarter-finals.
(Reporting by Julien Pretot, editing by Pritha Sarkar)
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