"It's wonderful to win in a Red Bull at the Red Bull Ring", said Verstappen, who was supported by an "Orange army" of more than 20,000 fans.
A furious Hamilton had already accused Mercedes of costing him victory after they failed to react to a Virtual Safety Car (VSC) period and stop their star driver for a tyre change.
Hamilton lost the lead of the Australian Grand Prix under due to a strategy miscalculation in March and the Mercedes pit wall also made a questionable call at the Chinese Grand Prix in April that saw Hamilton pass up an opportunity to mimic Daniel Ricciardo's race-winning strategy.
On a bleak afternoon for the champions, who had not been beaten at the Red Bull Ring since Austria returned to the calendar in 2014, Hamilton's team mate Valtteri Bottas retired with a mechanical failure. "Very consistent. No mistakes. So positive day but could have been better".
Hamilton slipped past pole-sitter Bottas in the first corner in an eventful start that also saw Raikkonen then almost snatch second from the Mercedes driver after the opening lap.
Hamilton then opened up a 2.4 seconds lead as the field was stretched on the tight, short track.
Kevin Magnussen, in the second Haas, took eighth place ahead of Renault pairing Carlos Sainz Jr and Nico Hulkenberg.
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Wolff also stressed that neither of the retirements were related to Mercedes's recent power unit upgrades.
His exit brought out the virtual safety vehicle, slowing down the pace and disallowing overtaking.
Hamilton was running in fourth, behind his rival Sebastian Vettel, before he then ran out of power with only eight laps remaining and stopped at the side of the track.
Mercedes head strategist James Vowles admitted to the error over the team radio and Hamilton said he went to console Vowles after his retirement. "I've got no time left on these tyres".
"The only ones who didn't have the tyre problems were the Ferraris, but all others suffered from heavy blistering and that was definitely not something that we expected", said Wolff.
It was clear that tyre wear was emerging as a critical factor.
On lap 53, he pitted again for super-softs, dropping a place to fifth behind Ricciardo, who nearly immediately slowed and stopped.
There might not have been any crashes during the race, but it was most certainly one of the most dramatic races of the season.