Lewis Hamilton was elevated to ninth by a post-race penalty for both Alfa Romeo drivers, having finished 11th despite leading much of the race from pole as his day fell apart amid a spin and costly penalty picked up when he dived into the pits after losing half his front wing.
That opened the door for Verstappen, who dominated the remainder of the race despite several more safety car restarts, to potentially put a foot in the door of the title race.
Charles Leclerc, Nico Hulkenberg and Valtteri Bottas all crashed out of contention at various stages of a totally crazy afternoon played out in the wet, with the Hockenheimring likely bowing out of F1 for now in fine style.
Verstappen kept himself in it, keeping hold of Hamilton throughout, despite five pit-stops and a spin of his own after taking on slick tyres, and was in position to capitalise when the champion's race suddenly fell apart.
Vettel's performance served as some redemption for last year's skate into the Sachskurve gravel, charging to second place in the final few laps after Bottas crashed, as the German also more than make up for Ferrari's qualifying nightmare.
Ferrari's day was darkened slightly by Leclerc spinning out, however.
Kvyat was a surprise podium finisher, as Toro Rosso took just their second ever top-three finish - and first since Vettel's Monza victory of 2008, another chaotic encounter.
The Russian, who became a father for the first time on Saturday night, has impressed on his return to F1 this year and the result could intensify rumours that he may replace Pierre Gasly, who retired late after another terrible race, which saw him overtaken numerous times by the Toro Rosso of Alexander Albon.
Lance Stroll was fourth for Racing Point, just missing out on a podium that would have been rich reward for a bold switch to slick tyres ahead of anyone late in the race.
The Canadian briefly led the race but just could not hang on to the top three.
Carlos Sainz picked up more good points in fifth, with Albon finishing a stupendous sixth, having driven an F1 car in the wet for the very first time.
The Alfa Romeos and the Haas cars came home in Noah's Ark order to round out the top 10, although Kimi Raikkonen and Antonio Giovinazzi were booted from the points post-race for a discrepancy found in their starting set-up.
The sanction gave Hamilton two bonus points, extending his championship lead to 41 points over Bottas and remarkably put Robert Kubica into the top 10 to score Williams' first point of a difficult campaign, as well as securing the Pole's first score in F1 since the 2010 Abu Dhabi GP.
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