The Dutch men then proved in a class of their own as they defended their team sprint title, leaving France trailing by almost a full second in the gold medal race.
Australia took the women's team sprint after a much closer duel against Russia with their duo of Kaarle McCulloch and Stephanie Morton striking gold.
Barker made her move just over two laps from the finish and outsprinted Dutchwoman Wild to win a fourth world title.
Seconds after the Welsh rider hit the front, a huge crash decimated the field as multiple riders tumbled to the boards, although there were no serious injuries reported.
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Belgium's Jolien D'Hoore took the bronze medal.
Barker's main success has come in Britain's women's team pursuit squad where she won gold at the Rio Oympics and was world champion in the discipline in 2013 and 2014.
She will have a chance to add to that haul on Thursday.
"I heard a bit of a bang so I didn't know the crash had happened until a lap later," said Barker.
"This was a big surprise because everything has been focused on the team pursuit."
The flying Dutch trio of Roy van den Berg, Harrie Lavreysen and Jeffrey Hoogland proved untouchable as they powered to the men's sprint gold, retaining the title they took on home boards in Apeldoorn last year.
Russia edged out Germany for men's bronze, although Germany's women, without sprint great Kristina Vogel who was paralysed in an accident last year, did take a bronze.
Thursday's action includes the men's team pursuit gold medal run between world and Olympic champions Britain and Australia.
There is also the women's team pursuit final and the men's 15km scratch race.
(Reporting by Martyn Herman; Editing by Ian Chadband)
